• Journal for History of Mathematics
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• v.17 no.1
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• pp.87-96
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• 2004

This article dealt with two approaches, algebraic and geometric approaches in terms of Pythagoreans theorem. As mathematics evolves, many theorems had been developed beginning with geometric approaches. However, the algebraic techniques that survive these days are so powerful and generalized in school curriculum. So, if students have more chances to see mathematical properties in geometrical ways, they can experience how beautiful and meaningful they are through the process of the advent of them. Also, it was to try to develop an insight into their applications to other problems.

• School Mathematics
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• v.5 no.3
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• pp.343-360
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• 2003

In this paper, we deal with the teaching of algebra based on patterns and generalization. The past algebra curriculum starts with letters(variables), algebraic expressions, and equations, but these formal approaching method has many difficulties in the school algebra. Therefore we insist the new algebraic approaches should be needed. In order to develop these instructions, we firstly investigate the relationship of patterns and algebra, the relationship of generalization and algebra, the steps of generalization from patterns and levels of difficulties. Next we look into the algebra instructions based arithmetic patterns, visual patterns and functional situations. We expect that these approaches help students learn algebra when they begin school algebra.

• Communications of Mathematical Education
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• v.25 no.2
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• pp.451-472
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• 2011

In this paper we studied problem solving related with geometric interpretation of algebraic expressions. We analyzed algebraic expressions, related these expressions with geometric interpretation. By using geometric interpretation we could find new approaches to solving mathematical problems. We suggested new problem solving methods related with geometric interpretation of algebraic expressions.

• School Mathematics
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• v.4 no.3
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• pp.347-360
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• 2002

This study analyzed the mathematical and didactical contexts of the Euclid's proof about Pythagorean theorem and compared with the teaching methods about Pythagorean theorem in school mathematics. Euclid's proof about Pythagorean theorem which does not use the algebraic methods provide students with the spatial intuition and the geometric thinking in school mathematics. Furthermore, it relates to various mathematical concepts including the cosine rule, the rotation, and the transfor-mation which preserve the area, and so forth. Visual demonstrations can help students analyze and explain mathematical relationship. Compared with Euclid's proof, Algebraic proof about Pythagorean theorem is very simple and it supplies the typical example which can give the relationship between algebraic and geometric representation. However since it does not include various spatial contexts, it forbid many students to understand Pythagorean theorem intuitively. Since both approaches have positive and negative aspects, reciprocal complementary role is required in pedagogical aspects.

• Journal of applied mathematics & informatics
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• v.30 no.5_6
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• pp.951-969
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• 2012

In this paper, we propose a class of meshless collocation approaches for the solution of time dependent partial differential algebraic equations (PDAEs) in terms of a radial basis function interpolation numerical scheme. Kansa's method and the Hermite collocation method (HCM) for PDAEs are given. A sensitivity analysis of the solutions from different shape parameter c is obtained by numerical experiments. With use of the random collocation points, we have obtain the more accurate solution by the methods than those by the finite difference method for the PDAEs with index-2, i.e, we avoid the influence from an index jump of PDAEs in some degree. Several numerical experiments show that the methods are efficient.

• The Transactions of the Korean Institute of Electrical Engineers
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• v.35 no.2
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• pp.77-84
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• 1986

The design of observers for the systems with unknown and unmeasurable disturbances is treated. A generalized observer design method is proposed and existence conditions are established by combining the existing two different approaches` disturbance modelling approach by O'Reilly and algebraic approach by Kudva et. al.. The proposed approach, therefore, takes the advantages and removes the shortcomings of the existing two approaches in view points of dimensionality and existence conditions of the observer. To show the usefulness of the approach, a numerical example is given.

• The Mathematical Education
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• v.41 no.1
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• pp.91-100
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• 2002

The purpose of this study is to help the students understand the concepts for the series and calculate the sum of series with diverse methods --- a) High School Algebraic Technique, b) Using Bernoulli Number, c) Integral Method, d) Euler Formular, e) Fourier Series

• Journal of Educational Research in Mathematics
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• v.20 no.3
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• pp.275-292
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• 2010

In this study, we discussed the way to teach algebra earlier through investigating to research trends of Early Algebra and researching about nature of subject involving algebra. There is a strong view that arithmetic and algebra have analogous forms and that algebra is on extension to arithmetic. Nevertheless, it is also possible to present a perspective that the fundamental goal and role of symbols and letters are difference between arithmetic and algebra. And, we could recognize that geometry was starting point of algebra trough historical perspectives. To consider these, we extracted some of possible directions to approaches to teach algebra earlier. To access to teaching algebra earlier, following ways are possible. (1) To consider informal strategy of young children. (2) Arithmetic reasoning considered of the algebraic relation. (3) Starting to algebraic reasoning in the context of geometrical problem situation. (4) To present young students to tool of letters and formular.

• Research in Mathematical Education
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• v.16 no.3
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• pp.177-194
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• 2012

What strategies are used to help students understand quadratic functions in mathematics classroom? In specific, how does Chinese teacher highlight a connection between algebraic representation and graphic representation? From October to November 2009, an experienced teacher classroom was observed. It was found that when students started learning a new type of quadratic function in lessons, the teacher used two different teaching strategies for their learning: (1) Eliciting students to plot the graphs of quadratic functions with pointwise approaches, and then construct the function image in their minds with global approaches; and (2) Presenting a specific mathematical problem, or introducing conception to elicit students to conjecture, and then encouraging them to verify it with appoint approaches.

• Journal of Korean Society for Quality Management
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• v.48 no.4
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• pp.535-552
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• 2020

Purpose: The main theme of this study is to derive a specific quality loss function with multiple characteristics according to the same analytical structure as the single characteristic quality loss function of Taguchi. In other words, it presents an analytical framework for measuring quality costs that can be controlled in practice. Methods: This study followed the analytical methodology through geometric, linear algebraic, and statistical approaches Results: The function suggested by this study is as follows; $$L(x_1,x_2,{\cdots},x_t)={\sum\limits_{i=1}^{t}}k_i\{x_i+{\sum\limits_{j=1}^{t}}\({\rho}_{ij}{\frac{d_i}{d_j}}\)x_j\}x_i$$ Conclusion: This paper derived the quality loss function with multiple quality characteristics to expand the usefulness of the Taguchi quality loss function. The function derived in this paper would be more meaningful to estimate quality costs under the practical situation and general structure with multiple quality characteristics than the function by linear algebraic approach in response surface analysis.