• East Asian mathematical journal
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• v.31 no.2
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• pp.167-188
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• 2015

Formula for the area of a trapezoid is an educational material that can handle algebraic and geometric perspectives simultaneously. In this note, we will make up the expression equivalent algebraically to the formula for the area of a trapezoid, and deal with the conversion of a geometric point of view, in algebraic terms of translating and interpreting the expression geometrically. As a result, the geometric conversion model, the first algebraic model, the second algebraic model are obtained. Therefore, this problem is a good material to understand the advantages and disadvantages of the algebraic and geometric perspectives and to improve the mathematical insight through complementary activity. In addition, these activities can be used as material for enrichment and gifted education, because it helps cultivate a rich perspective on diverse and creative thinking and mathematical concepts.

• Journal of Educational Research in Mathematics
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• v.12 no.2
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• pp.229-246
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• 2002

In this paper, we start with the question "what is algebraic thinking\ulcorner". The problem is that the algebraic thinking is not exactly defined. We consider algebraic thinking from the various perspectives. But in the discussion relating to the definition of algebraic thinking, we verify that there is the algebraic notations in the core of algebraic thinking. So we device algebraic notations into the six categories, and investigate these examples from the school mathematics. In order to investigate this relation of algebraic thinking and algebraic notations, we present 'the algebraic thinking process analysis model' from Frege' idea. In this model, there are three components of algebraic notations which interplays; sense, expression, denotapion. Thus many difficulties of algebraic thinking can be explained by this model. We suppose that the difficulty in the algebraic thinking may be caused by the discord of these three components. And through the transformation of conceptual frame, we can explain the dynamics of algebraic thinking. Also, we present examples which show these difficulties and dynamics of algebraic thinking. As a result of these analysis, we conclude that algebraic thinking can be explained through the semiotic aspects of algebraic notations.

• The Mathematical Education
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• v.41 no.3
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• pp.341-360
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• 2002

The purpose of the study was to investigate the effectiveness of dynamic software in solving high school analytic geometry problems compared with traditional algebraic approach. Three high school students who have revealed high performance in mathematics were involved in this study. It was considered that they mastered the basic concepts of equations of plane figure and curves of secondary degree. The research questions for the study were the followings: 1) In what degree students understand relationship between geometric approach and algebraic approach in solving geometry problems? 2) What are the difficulties students encounter in the process of using the dynamic software? 3) In what degree the constructions of geometric figures help students to understand the mathematical concepts? 4) What are the effects of dynamic software in constructing analytic geometry concepts? 5) In what degree students have developed the images of algebraic concepts? According to the results of the study, it was revealed that mathematical connections between geometric approach and algebraic approach was complementary. And the students revealed more rely on the algebraic expression over geometric figures in the process of solving geometry problems. The conceptual images of algebraic expression were not developed fully, and they blamed it upon the current college entrance examination system.

• Journal of Educational Research in Mathematics
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• v.24 no.3
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• pp.359-374
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• 2014

Students' difficulties and errors in relation to mathematical proofs are worth while to say one of the dilemmas in mathematics education. The potential elements of their difficulty are scattered over the process of proving in geometry as well as algebra. This study aims to investigate whether middle school students understand the context of algebraic proof including literal expressions. We applied 24 third-grade middle school students a test item which shows a proof including a literal expression and missing the conclusion. Over the half of them responded wrong answers based on only the literal expression without considering its context. Three of them were interviewed individually to show their thinking. As a result, we could find some characteristics of their thinking including the perspective on proof as checking the validity of algebraic expression and the gap between proving and understanding of proof etc. From these, we also discussed about several didactical implications.

• 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.

• Structural Engineering and Mechanics
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• v.76 no.3
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• pp.395-412
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• 2020

In this article, the values of internal force and deformation of a curved beam under any action with the firm or elastic supports are determined by using structural matrices. The article presents the general differential formulation of a curved beam in global coordinates, which is solved in an orderly manner using simple integrals, thus obtaining the transfer matrix expression. The matrix expression of rigidity is obtained through reordering operations on the transfer notation. The support conditions, firm or elastic, provide twelve equations. The objective of this article is the construction of the algebraic system of order twenty-four, twelve transfer equations and twelve support equations, which relates the values of internal force and deformation associated with the two ends of the directrix of the curved beam. This final algebraic system, expressed in matrix form, is divided into two subsystems: twelve algebraic equations of internal force and twelve algebraic equations of deformation. The internal force and deformation values for any point in the curved beam directrix are determined from these values in the initial position. The five examples presented show how to apply the matrix procedures developed in this article, whether they are curved beams with the firm or elastic support.

• Communications for Statistical Applications and Methods
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• v.3 no.3
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• pp.125-133
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• 1996

In this paper, we present a algebraic technique for computing system reliability for complex system. The method was originally developed as an aid to fault tree analysis but it applies to general problems of reliability assessment. A success expression which directly gives the reliability expression is formed and simplified by the procedure. Several algorithms and examples are illustrated.

• International Journal of Control, Automation, and Systems
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• v.1 no.4
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• pp.431-443
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• 2003

This paper presents an efficient method to evaluate the performance of an extended stochastic Petri net by simple algebraic operations. The reachability graph is derived from an extended stochastic Petri net, and then converted to a timed stochastic state machine, using a semi-Markov process. The n-th moments of the performance index are derived by algebraic manipulations with each of the n-th moments of transition time and transition probability. For the derivation, three reduction rules are introduced on the transition trajectories in a well-formed regular expression. Efficient computation algorithms are provided to automate the suggested method. The presented method provides a proficient means to derive both the numerical and the symbolic solutions for the performance of an extended stochastic Petri net by simple algebraic manipulations.

• The Mathematical Education
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• v.51 no.1
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• pp.47-61
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• 2012

Students can infer mathematical principles in a very natural way by connecting mutual relations between mathematical fields. These process can be revealed by taking tasks that can derive mathematical connections. The task of this study is to make expression and design it and derive mathematical principles from the design. This study classifies the mathematical field of expression for design and analyzes mathematical thinking process by connecting mathematical fields. To complete this study, 40 gifted students from 5 to 8 grade were divided into two classes and given 4 hours of instruction. This study analyzes their personal worksheets and e-mail interview. The students make expressions using a functional formula, remainder and figure. While investing mathematical principles, they generalized design by mathematical guesses, generalized principles by inference and accurized concept and design rules. This study proposes the class that can give the chance to infer mathematical principles by connecting mathematical fields by designing.

• Proceedings of the IEEK Conference
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• 2005.11a
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• pp.849-852
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• 2005

A factorization is an extremely important part of multi-level logic synthesis. The number of literals in a factored from is a good estimate of the complexity of a logic function, and can be translated directly into the number of transistors required for implementation. Factored forms are described as either algebraic or Boolean, according to the trade-off between run-time and optimization. A Boolean factored form contains fewer number of literals than an algebraic factored form. In this paper, we present a new method for a Boolean factorization. The key idea is to identify two-cube Boolean subexpression pairs from given expression. Experimental results on various benchmark circuits show the improvements in literal counts over the algebraic factorization based on Brayton's co-kernel cube matrix.