• Journal of Elementary Mathematics Education in Korea
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• v.24 no.1
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• pp.169-186
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• 2020

The purpose of this study is to find new material for developing teaching and learning materials for the gifted class of elementary school students by using the rotatable tangram made by modifying the traditional tangram. Rotatable tangram can be justified by gifted students through mathematical communication. However, even gifted class students have some limitations in finding and justifying triangles and rectangles of all sizes unless they go through the 'symbolization' stage at the elementary school level. Therefore, students who need an inquiry process for letters and symbols need to provide supplementary learning materials and additional questions. It is expected that the material of rotatable tangram for the development of teaching and learning materials for elementary school gifted students will contribute to the development of mathematical reasoning and mathematical communication ability.

• Education of Primary School Mathematics
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• v.25 no.2
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• pp.179-195
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• 2022

Mathematical communication competency, one of the six mathematical competencies emphasized in the latest mathematics curriculum, plays an important role both as a means and as a goal for students to learn mathematics. Therefore, it is meaningful to find instructional methods to improve students' mathematical communication competency and analyze their communication competency in detail. Given this background, this study analyzed 64 sixth graders' mathematical communication competency after they participated in the lessons of fraction division emphasizing mathematical communication. A written assessment for this study was developed with a focus on the four sub-elements of mathematical communication (i.e., understanding mathematical representations, developing and transforming mathematical representations, representing one's ideas, and understanding others' ideas). The results of this study showed that students could understand and represent the principle of fraction division in various mathematical representations. The students were more proficient in representing their ideas with mathematical expressions and solving them than doing with visual models. They could use appropriate mathematical terms and symbols in representing their ideas and understanding others' ideas. This paper closes with some implications on how to foster students' mathematical communication competency while teaching elementary mathematics.

• Lingua Humanitatis
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• v.6
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• pp.147-166
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• 2004

Whitehead's discussion of language is not to be found in any one book or article. It is interwoven with his discussion of many other questions. He was, however, greatly concerned with the problem of symbolism in general and the uses of language. He regards language, spoken or written, as an instrument devised by men to aid them in their adjustment to the environment in which they live Language is used for many specific purposes in the process of this adjustment. Words are employed not only to refer to data and to express emotions. They may be used also to record experiences, and thoughts about these experiences. Worts also function as instruments in the organization of experiences as they are considered in retrospect. Thus words free us from the bondage of the immediate. And Whitehead's theory of meaning is implicit in his discussion of the functions of language. According to him, the human mind is functioning symbolically when some components of its experience elicit consciousness, beliefs, emotions, and usages, respecting other components of its experiences. The former set of components are the 'symbols', and the latter set constitute the 'meaning' of the symbols. Whitehead points out that one word may have several meanings, i.e. refer to several different data. In order to understand, thus, the meaning to which a word refers, it is sometimes very important to appreciate the system of thought within which a person is operating. Further, Whitehead's discussion of language includes a number of cogent warning the deficiencies of language, and hence the need for great care in the use of words. In fact, language developed gradually. For the most part we have created words designed to deal with practical problems. Attention focuses on the prominent features in a situation, in particular the changing aspects of things. With reference to such data our words are relatively adequate. However, this issues in an unfortunate superficiality. The enduring, the subtle, the complex and the general aspects of the universe do not have adequate verbal representation. for this reason, Whitehead's position concerning the uses of language in speculative philosophy is stated with pungent directness. The uncritical trust in the adequacy of language is one of the main errors to which philosophy is liable. Since ordinary language does not do justice to the generalities, profundities and complexities of life, it is obvious that philosophy requires new words and phrases, or at least the revision of familiar words and phrases. Proceeding to develop the theme Whitehead contends that words and phrases must be stretched towards a generality foreign to their ordinary usage. In the same vein Whitehead refers to the need to realize that language which is the tool of philosophy needs to be redesigned just as in physical science available physical apparatus needs to be redesigned. But even these words and phrases, stretched or redesigned, are never completely adequate in philosophical speculations. They are, in his opinion, merely a great improvement over ordinary language or the language science, mathematics or symbolic logic.

• Journal of Elementary Mathematics Education in Korea
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• v.5 no.1
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• pp.99-120
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• 2001

The purpose of this thesis is to find the guiding direction of mathematical communication in lower grade students of elementary school and to present a new direction about the effect of using concrete material in communication. It is expected that mathematical communication increases when concrete material is used for the students of the lower grades, who are in concrete operational period. Therefore, this study ai s to investigate what characteristics there are in mathematical communication of second grade students and what effect concrete materials have on mathematical communication and learning. The analysis of the teaching record shows that the second grade students use alternative terms in the process of communication since they are not familiar with mathematical symbols or terms, which is a characteristic of communication in a mathematics class in which concrete material is used. In the process of teaming the students apply their living experiences to their teaming. Since a small number of students lead class, the interaction between students is also led by them. The direction of communication in a small group is not centered around solution of a problem, and most students show a more interest in finding answers than in the process of learning. The effect that concrete material has on communication plays an important role in promoting students' speaking activity; it allows students to identify and correct their errors more easily. It also makes students' activities more predictable, and it increases a small group activities through the medium of concrete material. However, it was also noticed that students' listening activities are not appropriately developed since they do not pay attention to a teacher who uses concrete material. The effects that concrete material has on mathematics class can be summarized as follows. Concrete material promotes students' participation in class by triggering their interest of learning of mathematics and helps them to understand the course of learning. It also helps the teaming and formation of concepts for children of low academic performance. And it makes a phased learning possible according to students' ability to use concrete material and to solve a problem. Based upon the results above mentioned, the use of concrete material is absolutely needed in mathematics classes of lower grade elementary school students since it increases communication and gives much influence on mathematics learning. Therefore, teachers need to develop teaching or learning method which can help increase communication, considering the characteristics of students' communication.

• Journal of Educational Research in Mathematics
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• v.23 no.2
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• pp.135-152
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• 2013

The purpose of this study is to navigate to the desired direction for the figure of number line through the extensive analysis of number line in middle school textbooks and literatures. For the efficient achievement of this purpose, three research questions were posed as follows: First, we compare the figures of number line in textbooks of Korea and other countries. Korean math textbooks mark the arrow on both sides of number line. But, however, coordinate plane was marked with arrow on only positive direction of number line. In contrast, the majority of secondary school textbooks in several foreign countries has the arrow only on positive direction. Second, the change in the figure of number line has been analyzed historically from two perspectives. From the first to 2007-revised curriculum, math textbooks of Korea were analyzed. Since the 6th curriculum, the number of textbooks with arrows on both sides has increased sharply. That is, textbooks with one arrow almost have disappeared. It is strange that any explanation for this abrupt change can't be found. The following analysis was also performed on published foreign literatures since Descartes. There was no arrow in the early figures of number line. But after 19th century, number lines with one arrow have begun to appear. Third, based on the previous study, we propose a reasonable way for the figure of number line. In fact, we claim that, in terms of linguistic symbols, the number line should be with only one arrow on positive side.

• The Mathematical Education
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• v.50 no.2
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• pp.213-231
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• 2011

The purpose of this study is to analyze the 6th graders' understanding of the concepts of variable on various aspects of school algebra. For this purpose, the test of concepts of variable targeting a sixth-grade class was conducted and then two students were selected for in-depth interview. The level of mathematics achievement of the two students was not significantly different but there were differences between them in terms of understanding about the concepts of variable. The results obtained in this study are as follows: First, the students had little basic understanding of the variables and they had many cognitive difficulties with respect to the variables. Second, the students were familiar with only the symbol '${\Box}$' not the other letters nor symbols. Third, students comprehended the variable as generalizers imperfectly. Fourth, the students' skill of operations between letters was below expectations and there was the student who omitted the mathematical sign in letter expressions including the mathematical sign such as x+3. Fifth, the students lacked the ability to reason the patterns inductively and symbolize them using variables. Sixth, in connection with the variables in functional relationships, the students were more familiar with the potential and discrete variation than practical and continuous variation. On the basis of the results, this study gives several implications related to the early algebra education, especially the teaching methods of variables.

• Journal of the Korean School Mathematics Society
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• v.13 no.1
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• pp.23-43
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• 2010

This study is to investigate how much highschool students, who have learned functional concepts included in the Middle school math curriculum, understand chapters of the function, to analyze the types of errors which they made in solving the mathematical problems and to look for the proper instructional program to prevent or minimize those ones. On the basis of the result of the above examination, it suggests a classification model for teaching-learning methods and teaching material development The result of this study is as follows. First, Students didn't fully understand the fundamental concept of function and they had tendency to approach the mathematical problems relying on their memory. Second, students got accustomed to conventional math problems too much, so they couldn't distinguish new types of mathematical problems from them sometimes and did faulty reasoning in the problem solving process. Finally, it was very common for students to make errors on calculation and to make technical errors in recognizing mathematical symbols in the problem solving process. When students fully understood the mathematical concepts including a definition of function and learned procedural knowledge of them by themselves, they did not repeat the same errors. Also, explaining the functional concept with a graph related to the function did facilitate their understanding,

• Journal of Educational Research in Mathematics
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• v.22 no.4
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• pp.619-636
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• 2012

The main purpose of this study was to explore the process of generalization generated by mathematically gifted students. Specifically, this study probed how fourth, fifth, and sixth graders might generalize geometric patterns and represent such generalization. The subjects of this study were a total of 30 students from gifted classes of one elementary school in Korea. The results of this study showed that on the question of the launch stage, students used a lot of recursive strategies that built mainly on a few specific numbers in the given pattern in order to decide the number of successive differences. On the question of the towards a working generalization stage, however, upper graders tend to use a contextual strategy of looking for a pattern or making an equation based on the given information. The more difficult task, more students used recursive strategies or concrete strategies such as drawing or skip-counting. On the question of the towards an explicit generalization stage, students tended to describe patterns linguistically. However, upper graders used more frequently algebraic representations (symbols or formulas) than lower graders did. This tendency was consistent with regard to the question of the towards a justification stage. This result implies that mathematically gifted students use similar strategies in the process of generalizing a geometric pattern but upper graders prefer to use algebraic representations to demonstrate their thinking process more concisely. As this study examines the strategies students use to generalize a geometric pattern, it can provoke discussion on what kinds of prompts may be useful to promote a generalization ability of gifted students and what sorts of teaching strategies are possible to move from linguistic representations to algebraic representations.

• Journal for History of Mathematics
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• v.21 no.4
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• pp.61-78
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• 2008

In this paper, we discuss about De Morgan's view on the development of algebra according to following distinctions: arithmetic, universal arithmetic, symbolic algebra, significant algebra. De Morgan thought that the differences between arithmetic and universal arithmetic lie in the usage of letters and the immediate performance of computation. In his viewpoint, universal arithmetic is a transitional phase, in which absurd phenomena occur, from arithmetic to algebra and these absurd phenomena call for algebra. The feature of De Morgan's view on the development of algebra is that symbolic calculus which consist of symbol system without symbol's meaning is acquired, then as extended meanings are furnished to symbols, symbolic calculus become logical so significant calculus is developed. For example, Single algebra is developed, as an extended meaning is furnished to a symbol -1, and double algebra is developed, as an extended meaning is furnished to a symbol $\sqrt{-1}$. According to De Morgan, a symbol system is derived from the incompleteness of a prior symbol system.

• The Journal of the Korea Contents Association
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• v.21 no.6
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• pp.435-444
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• 2021

This paper discusses the geometrical interpretation of the Taegeuk Shapes of Kameun Temple through the geometric analysis of mathematics. Based on the literature, This paper attempted to clarify that the origin of Gameunsa's founding of the spirit of patriotism may coincide with historical records through historical literature and geometric meaning. First, the background of the founding of Kameun temple, geographical location located near the East Sea, especially the history of the ancient Chinese mathematics at the time, And that mathematical knowledge influenced all fields such as agriculture, architecture, and art. Secondly, it is related to the historical record as the space of about 60 centimeters, which is uniquely underground, was identified as the structure of the excavated space. It is thought that there is a strong correlation with the origin that the King Munmu changed into a dragon, and set up the temple to be able to stay. Based on these, the clues of the interpretation of the taegeuk and the triangular pattern were searched in the samcheon yanggi(參天兩地) of the Oriental and circumference of the Western. The taegeuk and triangular patterns represent the symbols of yin-yang harmony, which correspond to the origin of its creation. the Korean people regarded the mysterious dragon as a symbol of yinyang harmony. In conclusion the Shapes of Kameun temple's stone is consistent with the contents mentioned in the historical record.