• Journal of Elementary Mathematics Education in Korea
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• v.20 no.3
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• pp.431-456
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• 2016

The purpose of this study is to investigate the appropriate time of introduction and the usage of the number line, in order to suggest the right point of learning the number concept to the elementary school students. For the efficient achievement of this purpose, we investigated the mathematical models for constructing the number concept such as number line, empty number line and double number line, counting and development of number concept. Then, we conducted case study on the time of introduction and the usage of the number line. Finally, we analyzed the result. First, there is need for adjustment to conduct the introduction of the number line from the second year of elementary school, so to help the students understand the continuing number concept through the understanding on the metaphorical concept of the number line. Second, there is the need of positive introduction and the use on the mathematical models; empty number line which helps to draw various thinking strategy visually through the process of operations such as addition and subtraction; the division into equal part and division by equal part in which multiplicative comparative situation or division takes place; the double number line which helps to understand the rate or proportional distribution. Finally, when adopting the number line, the empty number line, or the double number line, we suggested the necessity of learning about elaborate guidance and the usage in order to fully understand the metaphorical concept of the number line.

• Education of Primary School Mathematics
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• v.25 no.3
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• pp.213-232
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• 2022

With the importance of number line in learning fractions, this study investigated how 4th grade students understand fractions and its operations in number line. The questionnaire consisted 22 items which were related to representing fractions, comparing the size of fractions, and operating addition and subtraction of fractions. Both structured number line and sub-structured number line were presented in the items. As results of the study, the overall success rates were not high and even some items showed higher incorrect answer rates than the success rates. Also, the students showed a difficulty in solving non-structured number line tasks. It was also noticeable that students showed several types of incorrect answers, which means that students had incomplete understanding of both fractions and number line. This paper is expected to shed light on elementary students' understanding of fractions and number line and to provide implications of how to deal with number line in teaching and learning fractions in the elementary school.

• Journal of Educational Research in Mathematics
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• v.25 no.2
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• pp.189-206
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• 2015

There has been a recurring call for using visual representations in textbooks to improve the teaching and learning of proportional reasoning. However, the quantity as well as quality of visual representations used in textbooks is still very limited. In this article, we analyzed visual representations presented in a Grade 6 textbook from two perspectives of proportional reasoning, multiple-batches perspective and variable-parts perspective, and discussed the potential of the double number line and the double tape diagram to help develop the idea 'things covary while something stays the same', which is critical to reason proportionally. We also classified situations that require proportional reasoning into five categories and provided ways of using the double number line and the double tape diagram for each category.

• The Mathematical Education
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• v.61 no.3
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• pp.375-396
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• 2022

With the importance for teachers of understanding students' strategies and providing appropriate feedback to their students, the purpose of this study is to analyze how prospective elementary teachers interpret and respond students' strategies for fraction tasks with number lines. The findings from analysis of 64 prospective teachers' responses were as follow. First, the prospective teachers in general could identify the students' understanding and errors based on their strategies, however, some prospective teachers overgeneralized students' mathematical thinking at a superficial level. Second, the prospective teachers could pose diverse tasks or activities for revising the students' errors, while some prospective teachers tried to correct students' errors by using only the area models. Based on these results, this study suggests for prospective teachers to have opportunities to understand elementary students' diverse problem strategies and to consider teaching methods with different fraction models.

• Journal of Educational Research in Mathematics
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• v.27 no.4
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• pp.791-810
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• 2017

This study explored the possibility of using visual models in teaching proportional reasoning based on the review of previous studies. Many studies on proportional reasoning emphasize that students tend to simply apply formal procedures without understanding the meaning behind them and that using visual models may be an alternative to help students develop informal strategies and proportional reasoning. Given these, we re-constructed and implemented the unit of a textbook to teach sixth graders proportional reasoning with ratio table, double number line, and double tape diagram. The results of this study showed that such visual models helped students understand the meaning of proportion, explore the properties of proportion, and solve proportional problems. However, several difficulties that students experienced in using the visual models led us to suggest cautionary notes when to teach proportional reasoning with visual models. As such, this study is expected to provide empirical information for textbook developers and teachers who teach proportional reasoning with visual models.

• Journal of the korean Society of Automotive Engineers
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• v.5 no.4
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• pp.34-40
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• 1983

점탄성유체 유동에 대하여 일반적으로 U.C맥스웰 모델 또는 올드로이드 B모델이 사용되지만 이러한 모델들은-매우 큰 전단율 영역인 윤활문제에서 믿기 힘든-수직응력의 크기가 전단율의 제곱에 비례함을 나타내므로, 본 연구에서는 수직응력 계수들 (.psi.,.psi.$_{2}$)이 가정될 수 있는 Criminale-Ericksen-Filbey모델이 사용되었다. 2차 수직응력계수는 다른 문제들에서와 같이 무시되었으며 Weissenberg수가 포함된 특수레이놀즈식이 유도되었다. 이 모델의 속도분포는 -2차원 약한 점탄성 유체에 대하여 증명된 바와 같이-뉴우톤 유체와 같이 가정되었다. 유도된 특수레이놀즈 식은 Weissenberg수를 1까지 계산되었으며 그 결과 점탄성유체가 유한저어널 베 어링에서 유리한 것으로 나타났지만 그 차가 미소하여 일반베어링에서 점탄성 윤활유의 영향이 무시됨을 보였다.

• Journal of the Korean Institute of Illuminating and Electrical Installation Engineers
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• v.23 no.5
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• pp.58-65
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• 2009

A forest fire in the area of power line installations may be caused by flashover disturbances in power systems. In this study, experiments were conducted so as to investigate the reduction in dielectric strength caused by combustion flame, and flashover characteristics in the simulated condition of vertical-type model power lines were examined by making shorter and longer the horizontal distance(s) between combustion flame and high-voltage conductors under the application of 60[Hz] a.c. and d.c. high-voltages. As the results of the experimental investigation it is demonstrated that flame can reduce flashover voltages of the model air-gap in shorter range of the horizontal distance(s). Relative air density is considered in order to analyze the reduction causes of the flashover voltages, and it can be seen that the relative air density has a great influence on the flashover characteristics under the presence of combustion flame.

• The Bulletin of The Korean Astronomical Society
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• v.37 no.1
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• pp.102-102
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• 2012

타이탄의 근적외선 영역 스펙트럼에서 나타나는 특유의 넓은 흡수밴드를 고체 탄화수소화합물과 질소화합물의 조합으로 설명하고자 한다. 이 흡수 밴드는 대기권의 하층부에 존재하는 연무(haze)에 의한 것으로, 3.3 - 3.4 micron 부근과 2.30 - 2.35 micron 부근에서 유사한 형태가 발견됐다. 두 파장대 모두 C-H 스트레칭 모드가 존재하는 영역이라는 공통점이 있으며, 그 형태가 넓은 흡수밴드로 나타나므로 대기 중에 고체상태의 탄화수소화합물이 존재하는 것으로 유추할 수 있으며, 관련 분자들의 실험값과 복사전달모델을 이용하여 이를 설명하고자 한다. 또한, 이 흡수밴드는 고도에 따라 그 형태와 세기가 달라지므로, 연무 입자들의 고도에 따른 수직분포 및 크기 등을 파악할 수 있다. 이 연구에서는 기존에 타이탄에서 발견된 $CH_4$, $C_2H_6$, $CH_3CN$의 고체 분광선과, 해당 영역에서 흡수선을 보이는 $C_5H_{12}$, $C_6H_{12}$, $C_6H_{14}$ 등의 고체 분광선을 이용한 모델을 Cassini 탐사선의 VIMS 관측자료로부터 유도한 파장 및 고도에 따른 광학적 깊이 변화량과 비교하여 보여주고자 한다.

• Education of Primary School Mathematics
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• v.22 no.1
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• pp.29-48
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• 2019

Previous studies on double scale models, visual models with two different scales (Chong, 2015), have focused on double number line and little research has been conducted on how to employ double scale models in the elementary mathematics textbooks series. Given this, we analyzed the characteristics of double scale models in the Japanese elementary mathematics textbooks in the following aspects: (a) the contents of units where double scale models were used; (b) the purposes of using such models; (c) the types of such models tailored to the contents and grade levels; and (d) the characteristics of problem contexts dealing with the models. The results of this study showed that double scale models were effectively used to connect the contents related to multiplication, specifically for the contexts of ratio. Such models were addressed for students in a systematic and gradual way as the grade levels went up. Based on these results, this paper describes implications on how to use double scale models in mathematics textbooks.

• Journal of the Korean School Mathematics Society
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• v.11 no.1
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• pp.97-115
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• 2008

In high school mathematics class, to subtract a number b from a, we add the additive inverse of b to a and to divide a number a by a non-zero number b, we multiply a by the multiplicative inverse of b, which is the formal approach for operations of real numbers. This article aims to give a connection between the intuitive models in middle school mathematics class and the formal approach in high school for teaching operations of negative integers. First, we highlight the teaching methods(Hwang et al, 2008), by which subtraction of integers is denoted by addition of integers. From this methods and activities applying the counting model, we give new teaching methods for the rule that the product of negative integers is positive. The teaching methods with horizontal mathematization(Treffers, 1986; Freudenthal, 1991) of operations of integers, which is based on consistently applying the intuitive model(number line model, counting model), will remove the gap, which is exist in both teachers and students of middle and high school mathematics class. The above discussion is based on students' cognition that the number system in middle and high school and abstracted number system in abstract algebra course is formed by a conceptual structure.