• Journal of Educational Research in Mathematics
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• v.15 no.3
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• pp.287-313
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• 2005

The decimal fraction concept plays an important role in understanding the real number which is one of the major concepts in school mathematics. In the school mathematics of Korea, the decimal fraction is treated merely as a sort of name of the common fraction, while many other important aspects of the decimal fraction concept are ignored. In consequence students fail to understand the decimal fraction concept properly, and merely consider it as a kind of number for formal computation. Preceding studies also identified students' narrow understanding of the decimal fraction concept. But none of them succeeded in clarifying the essences of the decimal fraction concept, which are crucial for discussing the didactical problems of it. In this study we attempted a didactical analysis of the decimal fraction concept and disclosed the roots of didactical problems and presented measures for its improvement. First, we attempted a phenomenological analysis of the decimal fraction concept and extracted 9 elements of the decimal fraction concept. Second, we has analyzed of the essence of the decimal fraction concept more clearly by relating it to the situations where it functions and its representations. For this we tried to construct the conceptual field of the decimal fraction. Third, we categorized he developmental levels of the decimal fraction concept from the aspect of external manifestation of the internal order. On the basis of these results, we attempted hierarchical structuring of the elements of the decimal fraction concept. And using the results of such a didactical analysis on the decimal number concept we analyzed the mathematics curriculum and textbooks of our country, investigated levels of students' understanding of the decimal fraction concept, and disclosed related problems. Finally we suggested directions and measures for the improvement of teaching decimal fraction concept.

• 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.

• Education of Primary School Mathematics
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• v.1 no.1
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• pp.59-71
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• 1997

기하학적 사고력 개발이라는 우리의 목표는 궁극적으로 보다 낮은 수준의 학생들에게 보다 높은 수준으로 나아가게 하는 경험을 주는 것이다. 학생들이 보다 높은 수준에서 추론할 수 있도록 하기 위하여 그들이 보다 낮은 수준에서 충분하고 효율적인 학습 경험을 가져야 한다는 것이다. 예를 들면 분수에서 이루어지는 것처럼 기계적인 암기식으로 사물을 학습함으로써 수준(단계)을 뛰어 넘으려고 노력하면은 그들이 학습한 것에 관한 많은 것을 기억할 수 없을 것이다. 조작에 관한 보다 풍부한 경험과 시각적으로 입체감을 주는 설명을 들은 어린이들이 보다 훌륭한 공간 추론을 할 수 있을 것이라 믿는다. 본 고에서는 기하학적인 사고의 개발에 관한 van Hiele 모델이 초등학교에서 기하 수업의 토론을 위한 기초로서 사용되어졌다. 그 모델의 수준들이 묘사되었고 일반적으로 초등학교 아동들의 사고는 0수준과 1수준이라 는 것이 밝혀졌다. 단지 극소수의 아동들이 2수준의 사고에 도달해 있을 것이다. 그러나 만약 초등학교에서의 수업이 기하학적인 개념을 구성하는데 주안점을 둔다면 보다 많은 어린이들이 2 수준의 사고를 보여줄 수 있을 것으로 생각된다. 0 수준의 어린이들은 도형의 형태에 초점이 맞추어져있고 1 수준의 어린이들은 도형의 성질을 이해하는데 에 있다. 2 수준의 사고자는 도형의 포함관계를 이해하고 비공식적으로 추론 할 수 있다. 처음 세 수준에서의 활동들에 대한 지침이 주어져 있으며 0 수준과 1수준에 연관되는 다수의 활동들을 묘사했다. 0수준의 어린이들을 위해 묘사된 활동들은 그들이 2차원 및 3차원의 도형 둘 다를 시각화하는데 도움을 주는 것이다. 1 수준에서 사고하는 학습자들을 위해 묘사된 활동들은 2차원 및 3차원 도형의 성질들을 강조했다. 아울러 본 고에서 언급한 활동들은 상호교수에의 접근을 반영했다. 그러한 접근방식은 학습자들로 하여금 그들의 활동과 의견으로부터 개념을 구성하게 해주며 그들의 활동 결과에 대해 다른 사람들과 의사소통 함으로서 개념을 명확하게 다듬어지게 해줄 수 있을 것이다. 아울러 평가 활동들이 본고의 마지막 부분에 주어져있다. 그러한 활동들은 교사들에게 어린이들의 기하학적인 사고수준을 결정하게 해주며 학습자들로 하여금 수업시간 이외에 보다 높은 사고수준으로 나아가게 해줄 수 있을 것으로 기대된다.

• Journal of the Korean School Mathematics Society
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• v.25 no.4
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• pp.375-396
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• 2022

In this paper, to provide an idea for the 2022 revised mathematics curriculum by restructuring the content of the 2015 mathematics curriculum, the content elements of recurring decimals of textbooks, which showed differences in the curriculum of Korea and Japan, were analyzed. As a result of this study, in Korea, before the introduction of the concept of irrational numbers, repeating decimals were defined in the second year of middle school, and the relationship between repeating decimals and rational numbers was dealt with. In Japan, after studying irrational numbers in the third year of middle school, the terminology of repeating decimals is briefly dealt with. Then, when learning the concept of limit in the high school <Mathematics III> subject, the relationship between rational numbers and repeating decimals is dealt with. Based on the results of the study, in relation to the optimization of the amount of learning in the 2022 curriculum revision, implications for the introduction period of the circular decimal number, alternatives to the level of its content, and the teaching and learning methods were proposed.

• School Mathematics
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• v.19 no.2
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• pp.319-343
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• 2017

In this study, we hope to reveal specialized content knowledge(SCK) and its features necessary to analyze student's errors and difficulties about the concept of irrational numbers. The instruments and interview were administered to 3 in-service mathematics teachers with various education background and teaching experiments. The results of this study are as follows. First, specialized content knowledge(SCK) were characterized by the fixation to symbolic representation like roots when they analyzed the concentration and overlooking of the representations of irrational numbers. Secondly, we observed the centralization tendency on symbolic representation and the little attention to other representations as the standard of judgment about irrational numbers. Thirdly, In-service teachers were influenced by content of students' error when they analyzed the error and difficulties of students. Lately, we confirmed that the content knowledge about the viewpoint of procept and actual infinity of irrational numbers are most important during the analyzing process.

• Journal of Music and Human Behavior
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• v.3 no.1
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• pp.63-76
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• 2006

Melody is defined as adding spatial dimension to the rhythm which is temporal concept. Being able to understand melodic pattern and to reproduce the pattern also requires cognitive skills. Since 1980, there has been much research on the relationship between academic skills and music cognitive skills, and how to transfer the skills learned in music work to the academic learning. The study purported to examine various research outcomes dealing with the correlational and causal relationships between musical and academic skills. The two dominating theories explaining the connection between two skills ares are "neural theory" and "near transfer theory." The theories focus mainly on the transference of spatial and temporal reasoning which are reinforced in the musical learning. The study reviewed the existing meta-analysis studies, which provided evidence for positive correlation between academic and musical skills, and significance of musical learning in academic skills. The study further examined specific skills area that musical learning is correlated, such as mathematics and reading. The research stated that among many mathematical concepts, proportional topics have the strongest correlation with musical skills. Also with reading, temporal processing also has strong relationship with auditory skills and motor skills, and further affect language and literacy ability. The study suggest that skills learned in the musical work can be transferred to other areas of learning and structured music activities may be every efficient for children for facilitating academic concepts.

• Journal of Elementary Mathematics Education in Korea
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• v.12 no.1
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• pp.79-100
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• 2008

Freudenthal has advocated the reinvention method. In that method, the pupils start with a meaningful context, not ready-made concepts, and invent informative method through which he could arrive at the formative concepts progressively. In many face the reinvention method is contrary to the traditional method. In traditional method, which was named as 'concretization method' by Freudenthal, the pupils start with ready-made concepts, and applicate this concepts to various instances through which he could arrive at the understanding progressively. Through analysis, it turns out that Korea's seventh elementary mathematics textbook is based on concretization method. In this thesis, first of all, I will reconstruct probability unit of seventh elementary textbook according to Freudenthal's reinvention method. Next, I will perform teaching experiment which is ruled by new lesson design. Lastly, I analysed the effects of teaching experiment. Through this study, I obtained the following results and suggestions. First, the reinvention method is effective on the teaching of probability concept and algorithm. Second, in comparison with current textbook strand, my strand which made probability concept go ahead and combinatorics concept let behind is not deficiency. Third, tree diagram is effective matrix which contribute to formalization of combinatorics calculation. Lastly, except for fraction, diverse representation of probability, for example percentage or informal ratio expression must be introduced in teaching process.

• (The)Study of the Eastern Classic
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• no.36
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• pp.403-432
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• 2009

In this paper, I examined two things. First explore the of the reality(眞) in poems written on paintings and discussion on paintings in men of letters of Pre-chosun dynasty, furthemore, it analyzes the meaning and thought. To explain it, I would like to analyse the discourse surrounding the reality(眞) in poems written on paintings and discussion on paintings, and concept of the borrowing(假) and the unreal(幻). First of all, I examine the ways that the view of nature of Confucians is pertinent to the abiding in reverence and the investigation of principles of confucians, also painting is necessarry condition for their moral and spiritual self-cultivation. In the discourse surrounding the reality, it is to suggest from 'natural reality (天眞)' to escape from the reality(眞), and proceeds to examine the transformation of reality(眞) in mind(人心). Furthermore I examine the meaning of transform and delicate difference between borrowing(假) and the unreal(幻). In this process, concept of the borrowing(假) and the unreal(幻) is divided into the two meaning. False(假) is to become the Borrowing and painting, the unreal(幻) is change into the act of recognition. In conclusion, I examined the significant transformation of the reality(眞) and the unreal(幻). The reality(眞) has been recognized as a important concept, it is diverged into the Nature, outer things and painting mind, and the object of ultimate value in appreciation and painting of Pre-chosun dynasty.

• (The)Study of the Eastern Classic
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• no.56
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• pp.147-170
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• 2014

To divide by the concept of Cheon (天) before and after the period of Song Dynasty: before Song Dynasty; according to the ancient Book of Odes (Sigyeong-詩經), "Cheon (天) gives birth to a large number of people", and, Confucius(孔子) say "Cheon(天) gave me Virtue(德)." Mencius(孟子) say "The person done with all his heart knows Seong(性, personality), so if he knows such Seong(性, personality), then he knows Cheon(天)." In Doctrine of the Mean(中庸), it says "Cheon(天) ordered it to be called - Seong(性, personality)." So, Cheon(天) had a religious meaning, such as Sangje(上帝) - Supreme Ruler. During the Song period, Cheon(天), the source of its existence, had construed as Mugeuk i Taegeuk Non(無極而太極論 - Theory of Supreme Ultimate while being Indeterminate) and Theory of li and ki (iginon-理氣論). Juja (朱子, a honorary name of Juhui, 朱熹) had said a reasonable Cheon(天), that is, Heavenly Principle (天理 - Cheolli) by interpreting Cheon(天) as Taegeuk(太極 - Supreme Polarity) and li(理) of Muwi(無爲 - uncontrived action). That's why Juja had lost the religiosity because of his reasonable frame. The purpose of this dissertation is to identify of the quality of being religious of li(理) on the basis of attribute of Cheon(天) argued by Toegye and Juja. In the text of Seomyeong(西銘 - Western Inscription), we can see their interpretation of the content that Toegye as "西銘考證講義"(Lecture on Historical Research of Western Inscription), and Juja as "西銘解"(Commentary on the Western Inscription). Seomyeong(西銘 - Western Inscription) was expounded as a logic of 'iil bunsu' (理一分殊 - coherence is one and distinguished into many). '理一分殊' means to live in as meaningful as possible according to the human nature that has been bestowed upon thyself. Juja and Toegye both said that in the aspect of 'iil'(理一 - coherence is one), Reverence(事天) ought to be done, but to look into the aspect of 'bunsu'(分殊-distinguished into many), Juja argued that people should follow the order of Heavenly Principle(天理 - Cheolli), and Toegye argued that people should have to perform the filial piety(孝). There are differences in methods of Toegye and Juja on account of distinction between attributes of Cheon(天). Such a distinction affects the attribute of li(理). Juja said divisively that Soiyeon(所以然-why its principle is so) is li(理), and Sodangyeon(所當然-what should be so) is Sa(事-divine project). Toegye argued that Sodangyeon(所當然-what should be so) is indeed li(理). It is the position of Toegye that to know Seong(性-the personality) of Sodangyeon(所當然-what should be so) is the first, rather than to know Cheon(天) of Soiyeon(所以然-why its principle is so) that is out of reach in a faraway place. Seong(性-the personality) is li(理) that bestowed by Cheon(天). In view of discussion about the essence and existence, for Toegye, the existence is the first, rather than the essence. The issues of existence is now enabled to talk about amid the discussion of metaphysics, namely li(理). Different from Juja, a theory noticed in Toegye is the theory of 'Lijado'(理自到). 'Lijado'(理自到) denotes 'Li(理) leads on their own.' It tells that separate from thing-in-itself, there is an energy that moves and oversees the thing. This is an issue of response between "I" as the principal agent and other people. If "I" as the principal agent is sincere to others, the others will come to me insomuch as they will be revealed through me. Here, a problem between the host and guest arises. Toegye perceived this problem that do not see me and others as same, and also do not see me and others as two. This is the logic of 'ilii iiil'(一而二 二而一 - looks like one but two, looks like two but one) of '理一分殊' (coherence is one and distinguished into many). The first thing to do between these two processes is to recognize the existence of 'iil'(理一). Toegye strongly displays a religious attitude identifying Cheon(天)=Li (理)=Sangje(上帝- Supreme Ruler) in the same light.

• Journal of Educational Research in Mathematics
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• v.19 no.2
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• pp.307-319
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• 2009

Current school mathematics introduces addition/subtraction between natural numbers, fractions, decimal fractions, and square roots, step-by-step in order. It seems that, however, school mathematics focuses too much on learning the calculation method of addition/subtraction between each stages of numbers, to lead most of students to understand the coherent principle, lying in addition/subtraction algorithm between real numbers in all. This paper raises questions on this problematic approach of current school mathematics, in learning addition/subtraction. This paper intends to clarify the fact that, if we recognize addition/subtraction between numbers from the viewpoint of 'measuring' and 'common measure', as Dewey did when he argued that the psychological origin of the concept of number was measuring, then we could find some common principles of addition/subtraction operation, beyond the superficial differences among algorithms of addition/subtraction between each stages of numbers. At the end, this paper suggests the necessity of improving the methods of learning addition/subtraction in current school mathematics.