• Journal of Educational Research in Mathematics
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• v.12 no.4
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• pp.457-472
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• 2002

In this paper, we deal with the algebraic thinking from the perspective of ‘process’ and ‘object’ aspects. Generally, mathematical concepts have come from the concrete process. We consider the origin of algebra as the arithmetic calculations. Also, the concept of school arithmetic is beginning from actions or procedures. However, in order to develop the alge- braic thinking and to apply this thinking, we have to see the history of algebraic thinking, and find this duality. Next we investigate various researches relating to the ‘process-object duality’. Theses studies suppose that the concept formation and thinking process should be stared from the process-object duality. Finally, we reinterprete many difficulties in algebra - equals sign, variables, algebraic expressions, and linear equations, the principle of permanence of form- from the perspective of the process-object duality.

• Journal of the Korea Academia-Industrial cooperation Society
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• v.10 no.11
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• pp.3227-3233
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• 2009

A method for performing Boolean resubstitution is proposed. This method is efficiently implemented using division matrix. It begins by creating an algebraic division matrix from given two logic expressions. By introducing Boolean properties and adding literals into the algebraic division matrix, we make the Boolean division matrix. Using this extended division matrix, Boolean substituted expressions are found. Experimental results show the improvements in the literal counts over well-known logic synthesis tools for some benchmark circuits.

• Journal of Communications and Networks
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• v.18 no.2
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• pp.182-189
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• 2016

The focus in this paper is on obtaining tight, simple algebraic-form bounds and invertible expressions for the average symbol error probability (ASEP) of M-ary phase shift keying (MPSK) in a class of composite fading channels. We employ the mixture gamma (MG) distribution to approximate the signal-to-noise ratio (SNR) distributions of fading models, which include Nakagami-m, Generalized-K ($K_G$), and Nakagami-lognormal fading as specific examples. Our approach involves using the tight upper and lower bounds that we recently derived on the Gaussian Q-function, which can easily be averaged over the general MG distribution. First, algebraic-form upper bounds are derived on the ASEP of MPSK for M > 2, based on the union upper bound on the symbol error probability (SEP) of MPSK in additive white Gaussian noise (AWGN) given by a single Gaussian Q-function. By comparison with the exact ASEP results obtained by numerical integration, we show that these upper bounds are extremely tight for all SNR values of practical interest. These bounds can be employed as accurate approximations that are invertible for high SNR. For the special case of binary phase shift keying (BPSK) (M = 2), where the exact SEP in the AWGN channel is given as one Gaussian Q-function, upper and lower bounds on the exact ASEP are obtained. The bounds can be made arbitrarily tight by adjusting the parameters in our Gaussian bounds. The average of the upper and lower bounds gives a very accurate approximation of the exact ASEP. Moreover, the arbitrarily accurate approximations for all three of the fading models we consider become invertible for reasonably high SNR.

• Communications of Mathematical Education
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• v.22 no.2
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• pp.137-164
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• 2008

Given the importance of early experience in algebraic thinking, we designed six consecutive lessons in which $4^{th}$ graders were encouraged to recognize patterns in the process of finding the relationships between two quantities and to represent a given problem with various mathematical models. The results showed that students were able to recognize patterns through concrete activities with manipulative materials and employ various mathematical models to represent a given problem situation. While students were able to represent a problem situation with algebraic expressions, they had difficulties in using the equal sign and letters for the unknown value while they attempted to generalize a pattern. This paper concludes with some implications on how to connect algebraic thinking with students' arithmetic or informal thinking in a meaningful way, and how to approach algebra at the elementary school level.

• 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.

• The Mathematical Education
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• v.44 no.4 s.111
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• pp.525-534
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• 2005

The purpose of this study is to analyze difficulties in the density word problem solving process of seventh graders and to search for the way to increase their problem solving ability in the density word problem. The results of this study could help teachers diagnose students' difficulties involved in density word problem and remedy the understanding of the concept of density, algebraic expressions, and algebraic symbols.

• Research in Mathematical Education
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• v.14 no.1
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• pp.33-50
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• 2010

The mathematical thinking which transforms important mathematical content and developed into mathematical structure is a vital process in building up mathematical ability as mathematical knowledge based on structure. Such process based on students' recognition of mathematical concept. Developing mathematical thinking into mathematical structure happens when different cognitive units are connected and compressed to form schema of solution, which could happen through some guided problems. The effort of arithmetic approach in problem solving did not necessarily provide students the structure schema of solution. The using of equation to solve the problem is based on the schema of building equation, and is not necessary recognizing the structure of the solution, as the recognition of structure may be lost in the process of simplification of algebraic expressions, leaving only the final numeric answer of the problem.

• East Asian mathematical journal
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• v.37 no.2
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• pp.265-288
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• 2021

In the 2015 revised mathematics curriculum, the system of equations is first introduced in 'Variables and Expressions' of [Middle School Grades 1-3]. Then, It is constructed that after learning the linear function in 'Functions', the relationship between the graphs of two linear functions and the systems of linear equations are learned so that students could improve the geometric representation of the systems of equations. However, in of Elective-Centered Curriculum Common Courses, Instruction is limited to algebraic manipulation when teaching and learning systems of quadratic equations. This paper presented the teaching and learning method that can improve students' mathematical connection through various representations by providing geometric representations in parallel using GeoGebra, a mathematics learning software, with algebraic solutions in the teaching and learning situation of simultaneous quadratic equations.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.14 no.4
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• pp.49-59
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• 2004

Boolean function synthesize problem is to find a boolean expression with in/outputs of original function. This problem can be modeled into a 0-1 integer programming. In this paper, we find a boolean expressions of S-boxes of DES for an example, whose algebraic structure has been unknown for many years. The results of this paper can be used for efficient hardware implementation of a function and cryptanalysis using algebraic structure of a block cipher.

• Journal of Gifted/Talented Education
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• v.26 no.1
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• pp.211-230
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• 2016

The study aimed to investigate the characteristics of algebraic thinking of the mathematically gifted students and search for how to teach algebraic thinking. Research subjects in this study included 93 students who applied for a science gifted education center affiliated with a university in 2015 and previously experienced gifted education. Students' responses on an algebraic item of a creative thinking test in mathematics, which was given as screening process for admission were collected as data. A framework of algebraic thinking factors were extracted from literature review and utilized for data analysis. It was found that students showed difficulty in quantitative reasoning between two quantities and tendency to find solutions regarding equations as problem solving tools. In this process, students tended to concentrate variables on unknown place holders and to had difficulty understanding various meanings of variables. Some of students generated errors about algebraic concepts. In conclusions, it is recommended that functional thinking including such as generalizing and reasoning the relation among changing quantities is extended, procedural as well as structural aspects of algebraic expressions are emphasized, various situations to learn variables are given, and activities constructing variables on their own are strengthened for improving gifted students' learning and teaching algebra.