• Journal for History of Mathematics
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• v.21 no.1
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• pp.109-120
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• 2008

Today linear algebra is one of compulsory courses for university mathematics by virtue of its theoretical fundamentals and fruitful applications. However, a mechanical computation-oriented instruction or a formal concept-oriented instruction is difficult and dull for most students. In this context, how to teach mathematical concepts successfully is a very serious problem. As a solution for this problem, we suggest establishing original concepts in linear algebra from the students' point of view. Any original concept means not only a practical beginning for the historical order and theoretical system but also plays a role of seed which can build most of all the important concepts. Indeed, linear algebra has exactly two original concepts : geometry of planes, spaces and linear equations. The former was investigated in [2], the latter in the present paper.

• School Mathematics
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• v.9 no.2
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• pp.197-222
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• 2007

The study aims to reflect the elementary school geometry education based on the Realistic Mathematics Education in the Netherlands in the light of the results from recent researches in geometry education and the direction of geometry standards for school mathematics of the National Council of Teachers of Mathematics in order to induce implications for improving korean geometry curriculum and textbook series. In order to attain these purposes, the present paper reflects the history of elementary school geometry education in the Netherlands, sketches the elementary school geometry education based on the Realistic Mathematics Education in the Netherlands by reflecting general goals of the mathematics education, the core goals for geometry strand of the Netherlands, and geometry and spatial orientation strand of Dutch Pluspunt textbook series for the elementary school more concretely. Under these reflections on the documents, it is analyzed what is the characteristics of geometry strand in the Netherlands as follows: emphasis on realistic spatial phenomenon, intuitive and informal approach, progressive approach from intuitive activity to spatial reasoning, intertwinement of mathematics strands and other disciplines, emphasis on interaction of the students, cyclical repetition of experiencing phase, explaining phases, and connecting phase. Finally, discussing points for improving our elementary school geometry curriculum and textbook series development are described as follows: introducing spatial orientation and emphasizing spatial visualization and spatial reasoning with respect to the instruction contents, considering balancing between approach stressing on grasping space and approach stressing on logical structure of geometry, intuitive approach, and integrating mathematics strands and other disciplines with respect to the instruction method.

• 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.

• Communications of Mathematical Education
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• v.23 no.3
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• pp.643-682
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• 2009

There is no doubt that the national competitiveness, in 21st century, definitely depends on how effectively it has been producing high-qualify human resources. It is inevitable that we are required to produce outstanding people who are going to make the use of highly developed scientific technology. Every country has already set to concentrate their all efforts in cultivating competitive human resources, enabling it to strengthen its national competitiveness. We Korea, in order to keep up with it, have arranged legal and systematic basis and are putting spurs to producing competent human resources under the 영재교육진흥법 및 시행령, which took effect from March, 2002. With the lack of experience and short history of Gifted Education, however, it is true that we still have many problems in promoting it in reality, We are asked to improve it by finding out what problems we have in whole area of Gifted Education, such as defining conception, choosing target students, structuring system and managing students afterwards. Therefore, this study, especially focusing on Math of Gifted Education is to investigate the present situation of Gifted Education and to examine what we should do for administering Gifted Education in effective ways.

• Communications of Mathematical Education
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• v.20 no.2 s.26
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• pp.295-326
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• 2006

Despite the importance of mathematics education, many students in high school have lost their interests and felt difficulties and they don't have 'mathematical' experience with meanings attached because of the entrance examination. This paper attempted to resolve these problems and find the teaching-method with which students can study by themselves with more confidence. Nowadays students' use of Internet is very popular. After develop 'the development of mathematical power' program based on mathematics history, history, science, the application of problems in real world, and self-evaluation, I made students repeat them after making teaching lessons in classroom as moving pictures. Through this processes, I attempted to develop the Self-Directed Learning' ability by making public education substantial. First of all I analyzed the actual conditions on 'Self-Directed Learning' ability in mathematics subject, the conditions of seeing and hearing in Internet learning program, and students' and their parents' interests in Internet education. By analyzing the records, I observed the significance of the introducing mathematics history in mathematics subject in early stager, cooperative-learning, leveled-learning, self-directed learning, and Internet learning. Actually in aspect of applying 'the development of mathematical power' program, at first I made up the educational conditions to fix the program, collected the teaching materials, established the system of teaching-learning model, developed materials for the learning applying Internet mail and instruments of classroom, and carried out instruction to establish and practice mathematics learning plan. Then I applied the teaching-learning model of leveled cooperation and presentation loaming and at the same time constructed and used the leveled learning materials of complementary, average, and advanced process and instructed to watch teaching moving pictures through Internet mail and in the classroom. After that I observed how effective this program was through the interest arid attitude toward mathematics subject, learning accomplishment, and the change of self-directed learning. Finally, I wrote the conclusion and suggestion on the preparation of conditions fur the students' voluntary participation in mathematics learning and the project and application on 'the development of mathematical power' program and repeated learning with the materials of moving pictures in classroom.

• Journal for History of Mathematics
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• v.23 no.3
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• pp.21-31
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• 2010

A latin square of order n is an $n{\times}n$ array with entries from a set of n numbers arrange in such a way that each number occurs exactly once in each row and exactly once in each column. Two latin squares of the same order are orthogonal latin square if the two latin squares are superimposed, then the $n^2$ cells contain each pair consisting of a number from the first square and a number from the second. In Europe, Orthogonal Latin squares are the mathematical concepts attributed to Euler. However, an Euler square of order nine was already in existence prior to Euler in Korea. It appeared in the monograph Koo-Soo-Ryak written by Choi Seok-Jeong(1646-1715). He construct a magic square by using two orthogonal latin squares for the first time in the world. In this paper, we explain Choi' s orthogonal latin squares and the history of the Orthogonal Latin squares.

• Journal for History of Mathematics
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• v.28 no.2
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• pp.85-102
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• 2015

Elliptic curves are a common theme among various fields of mathematics, such as number theory, algebraic geometry, complex analysis, cryptography, and mathematical physics. In the history of elliptic curves, we can find number theoretic problems on the one hand, and complex function theoretic ones on the other. The elliptic curve theory is a synthesis of those two indeed. As an overview of the history of elliptic curves, we survey the Diophantine equations of 3rd degree and the congruent number problem as some of number theoretic trails of elliptic curves. We discuss elliptic integrals and elliptic functions, from which we get a glimpse of idea where the name 'elliptic curve' came from. We explain how the solution of Diophantine equations of 3rd degree and elliptic functions are related. Finally we outline the BSD conjecture, one of the 7 millennium problems proposed by the Clay Math Institute, as an important problem concerning elliptic curves.

• Journal for History of Mathematics
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• v.19 no.4
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• pp.117-132
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• 2006

For a meaningful learning of the Construction Unit in 7-B Grade, this study aims to develop teaming materials on the basis of Clairaut's $El{\`{e}}ments$ de $G{\`{e}}om{\`{e}}trie$, which is grounded on a natural generation derived from the history of mathematics and emphasizes students' inquiry activity and reflective thinking activity, and to analyze the characteristics of learning process shown in classes which use the application of teaming materials. Six students were sampled by gender and performance and an interpretive case study was conducted. Construction was specified so as to be consciously executed with emphasis on an analysis to enable one to discover construction techniques for oneself from a standpoint of problem solving, a justification to reveal the validity of construction, and a step of reflection to generalize the results of construction.

• 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.27 no.3
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• pp.197-210
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• 2014

Species like insects and fishes have, in many cases, non-overlapping time intervals of one generation and their descendant one. So the population dynamics of such species can be formulated as discrete models. In this paper various discrete population models are introduced in chronological order. The author's investigation starts with the Malthusian model suggested in 1798, and continues through Verhulst model(the discrete logistic model), Ricker model, the Beverton-Holt stock-recruitment model, Shep-herd model, Hassell model and Sigmoid type Beverton-Holt model. We discuss the mathematical and practical significance of each model and analyze its properties. Also the stability properties of stationary solutions of the models are studied analytically and illustratively using GSP, a computer software. The visual outputs generated by GSP are compared with the analytical stability results.