• Journal of Educational Research in Mathematics
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• v.17 no.4
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• pp.453-475
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• 2007

School algebra starts with introducing algebraic expressions which have been one of the cognitive obstacles to the students in the transfer from arithmetic to algebra. In the recent studies on the teaching school algebra, algebraic thinking is getting much more attention together with algebraic expressions. In this paper, we examined the processes of the transfer from arithmetic to algebra and ways for teaching early algebra through algebraic thinking factors. Issues about algebraic thinking have continued since 1980's. But the theoretic foundations for algebraic thinking have not been founded in the previous studies. In this paper, we analyzed the algebraic thinking in school algebra from historico-genetic, epistemological, and symbolic-linguistic points of view, and identified algebraic thinking factors, i.e. the principle of permanence of formal laws, the concept of variable, quantitative reasoning, algebraic interpretation - constructing algebraic expressions, trans formational reasoning - changing algebraic expressions, operational senses - operating algebraic expressions, substitution, etc. We also identified these algebraic thinking factors through analyzing mathematics textbooks of elementary and middle school, and showed the middle school students' low achievement relating to these factors through the algebraic thinking ability test. Based upon these analyses, we argued that the readiness for algebra learning should be made through the processes including algebraic thinking factors in the elementary school and that the transfer from arithmetic to algebra should be accomplished naturally through the pre-algebra course. And we searched for alternative ways to improve algebra curriculums, emphasizing algebraic thinking factors. In summary, we identified the problems of school algebra relating to the transfer from arithmetic to algebra with the problem of teaching algebraic thinking and analyzed the algebraic thinking factors of school algebra, and searched for alternative ways for improving the transfer from arithmetic to algebra and the teaching of early algebra.

• Education of Primary School Mathematics
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• v.19 no.1
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• pp.1-18
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• 2016

Patterns are of great significance to develop algebraic thinking of elementary students. This study analyzed teaching methods of patterns in current elementary mathematics textbook series in terms of three main activities related to pattern generalization (i.e., analyzing the structure of patterns, investigating the relationship between two variables, and reasoning and representing the generalized rules). The results of this study showed that such activities to analyze the structure of patterns are not explicitly considered in the textbooks, whereas those to explore the relationship between two variables in a pattern are emphasized throughout all grade levels using function table. The activities to reason and represent the generalized rules of patterns are dealt in a way both for lower grade students to use informal representations and for upper grade students to employ formal representations with expressions or symbols. The results of this study also illustrated that patterns in the textbooks are treated rather as a separate strand than as something connected to other content strands. This paper closes with several implications to teach patterns in a way to foster early algebraic thinking of elementary school students.

• Journal for History of Mathematics
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• v.17 no.3
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• pp.33-42
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• 2004

${\varrho}$ is the real constant number that appears not only in calculus but also in a real life. The concept of the number ${\varrho}$ first appeared in an appendix of Napier's work on logarithms in 1618. The early developments on the logarithm became part of an understanding of the number ${\varrho}$. In 1727, the number ${\varrho}$ was studied by Euler explicitly. It ton14 almost 100 years to understand the number ${\varrho}$ which we learn in high school nowadays. By studying the origin of the number ${\varrho}$, we can guess that many mathemetician's research in our time will have significant meaning in the future although it looks like just some calculations of cohomology or K-theory etc.

• Journal for History of Mathematics
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• v.17 no.4
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• pp.27-36
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• 2004

This paper addresses G$\ddot{o}$del's critique of Turing's mechanism that a configuration of the Turing machine corresponds to each state of human mind. The first part gives a quick overview of Turing's analysis of cognition as computation and its variants. In the following part, we describe the concept of Turing machines, and the third part explains the computational limitations of Turing machines as a cognitive system. The fourth part demonstrates that Godel did not agree with Turing's argument, sometimes referred to as mechanism. Finally, we discuss an oracle Turing machine and its implications.

• Journal of Elementary Mathematics Education in Korea
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• v.15 no.2
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• pp.247-282
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• 2011

In the elementary school mathematics textbooks of the 7th national curriculum, just simple construction education is provided by having students draw a circle and triangle with compasses and drawing vertical and parallel lines with a set square. The purpose of this study was to examine the mathematical thinking of sixth-grade elementary school students in the construction process in a bid to give some suggestions on elementary construction guidance. As a result of teaching the sixth graders in gifted and nongifted classes about the equal division of line segments and evaluating their mathematical thinking, the following conclusion was reached, and there are some suggestions about that education: First, the sixth graders in the gifted classes were excellent enough to do mathematical thinking such as analogical thinking, deductive thinking, developmental thinking, generalizing thinking and symbolizing thinking when they learned to divide line segments equally and were given proper advice from their teacher. Second, the students who solved the problems without any advice or hint from the teacher didn't necessarily do lots of mathematical thinking. Third, tough construction such as the equal division of line segments was elusive for the students in the nongifted class, but it's possible for them to learn how to draw a perpendicular at midpoint, quadrangle or rhombus and extend a line by using compasses, which are more enriched construction that what's required by the current curriculum. Fourth, the students in the gifted and nongifted classes schematized the problems and symbolized the components and problem-solving process of the problems when they received process of the proble. Since they the urally got to use signs to explain their construction process, construction education could provide a good opportunity for sixth-grade students to make use of signs.

• 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.

• The KIPS Transactions:PartB
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• v.9B no.3
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• pp.277-286
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• 2002

Genetic programming (GP) has been combined with quantum mechanical perturbation theory to make a new algorithm to construct mathematical models and perform predictions for chaotic time series from real world. Procedural similarities between time series modeling and perturbation theory to solve quantum mechanical wave equations are discussed, and the exemplary GP approach for implementing them is proposed. The approach is based on multiple populations and uses orthogonal functions for GP function set. GP is applied to original time series to get the first mathematical model. Numerical values of the model are subtracted from the original time series data to form a residual time series which is again subject to GP modeling procedure. The process is repeated until predetermined terminating conditions are met. The algorithm has been successfully applied to construct highly effective mathematical models for many real world chaotic time series. Comparisons with other methodologies and topics for further study are also introduced.

• Journal of the Korean School Mathematics Society
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• v.19 no.1
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• pp.63-82
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• 2016

Mathematical notation is the main means to realize the power of mathematics. Under this perspective, this study analyzed the difficulties of learning an irrational number concept in terms of notation. I tried to find ways to overcome the difficulties arising from the notation. There are two primary ideas in the notation of irrational number using root. The first is that an irrational number should be represented by letter because it can not be expressed by decimal or fraction. The second is that $\sqrt{2}$ is a notation added the number in order to highlight the features that it can be 2 when it is squared. However it is difficult for learner to notice the reasons for using the root because the textbook does not provide the opportunity to discover. Furthermore, the reduction of the transparency for the letter in the development of history is more difficult to access from the conceptual aspects. Thus 'epistemological obstacles resulting from the double context' and 'epistemological obstacles originated by strengthening the transparency of the number' is expected. To overcome such epistemological obstacles, it is necessary to premise 'providing opportunities for development of notation' and 'an experience using the notation enhanced the transparency of the letter that the existing'. Based on these principles, this study proposed a plan consisting of six steps.

• School Mathematics
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• v.5 no.4
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• pp.477-506
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• 2003

In this paper, we analyze nine-grade students' conceptions of parameters, their relation to unknowns and variables and the process of understanding of letters in problem solving of equations and functions. The roles of letters become different according to the letters-used contexts and the meaning of letters Is changed in the process of being used. But, students do not understand the meaning of letters correctly, especially that of parameter. As a result, students operate letters in algebraic expressions according to the syntax without understanding the distinction between the roles. Therefore, the parameter of learning should focus on the dynamic change of roles and the flexible thinking of using letters. We develop a self-regulation model based on the monitoring working question in teaching-learning situations. We expect that this model helps students understand concepts of letters that enable to construct meaning in a concrete context.

• Journal of Educational Research in Mathematics
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• v.19 no.1
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• pp.81-98
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• 2009

This study analyzed the types of quantitative reasoning and the characteristics of representation in order to figure out the characteristics of quantitative reasoning of the sixth graders. Three students who used quantitative reasoning in solving problems were interviewed in depth. Results showed that the three students used two types of quantitative reasoning, that is difference reasoning and multiplicative reasoning. They used qualitatively different quantitative reasoning, which had a great impact on their problem-solving strategy. Students used symbolic, linguistic and visual representations. Particularly, they used visual representations to represent quantities and relations between quantities included in the problem situation, and to deduce a new relation between quantities. This result implies that visual representation plays a prominent role in quantitative reasoning. This paper included several implications on quantitative reasoning and quantitative approach related to early algebra education.