• Communications of Mathematical Education
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• v.24 no.2
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• pp.345-363
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• 2010

The purpose of this study is to investigate and analyze informal knowledge of students who do not learn the conception of proportion and to identify how the informal knowledge can be used for teaching the conception of proportion in order to present an effective method of teaching the conception. For doing this, proportion was classified into direct and inverse proportion, and 'What are the informal knowledge of students?' were researched. The subjects of this study were 117 sixth-graders who did not have prior learning on direct and inverse proportion. A total eleven problems including seven for direct proportion and four for inverse proportion, all of them related to daily life. The result are as follows; Even though students didn't learn about proportion, they solve the problems of proportion using informal knowledge such as multiplicative reasoning, proportion reasoning, single-unit strategy etc. This result implies mathematics education emphasizes student's informal knowledge for improving their mathematical ability.

• 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.

• Journal of the Korean School Mathematics Society
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• v.11 no.4
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• pp.609-627
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• 2008

The purpose of this research was to investigate the relationship between the students' strategy types and the recognition for proportional situations. The students' strategy types which were based on the results of ratio and proportion tests were divided into an additive type, a multiplicative type, and a formal type. This research analyzed the students' activities of categorization when were given the proportional problems and nonproportional problems to the students. And it also explored how to develop students' recognizing for the discrimination between the proportional situations and nonproportional situations. The results was the following. First, the students didn't discriminate the proportional situations and the nonproportional situations in the initial state but they came to discriminate little by little. Secondly, the students didn't discriminate the direct proportions and the inverse proportions until the last stage. Third, the multiplicative type was outperformed more than the formal type in solving the ratio and proportion problems but the formal type was outperformed more than the multiplicative type in discriminating between proportional situations and nonproportional situations. These results are interpreted as showing that solving ratio and proportion tasks and recognizing proportional situations are different aspects of proportional reasoning and it is necessary to understand multiplicative strategy with formal strategy in recognizing proportional situations.

• Journal of Educational Research in Mathematics
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• v.17 no.4
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• pp.359-380
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• 2007

This study was conducted with two 6-graders to identify how were their proportional reasoning abilities, whether they evolved proportional thinking in a various context, and what had influence on their proportional thinking. The findings, as previous researches noted, suggested that the proportional expression obtaining by instrumental understanding could not provide rich opportunities for students to improve understanding about ratio and proportion and proportional reasoning abilities, while being useful for determining the answers. The students were able to solve proportional problems with incorporating their knowledge of divisor, multiples, and fraction into proportional situations, but not the lack of number sense. The students easily solved proportional problems experienced in math and other subjects but they did not notice proposition in problems with unfamiliar contexts.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.21 no.2
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• pp.199-208
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• 2008

In this paper, to analyze the initial valued non-homogeneous elastic half space by the scaled boundary analysis, the infinite element approach was introduced. The free surface of the initial valued non-homogeneous elastic half space was modeled as a circumferential direction of boundary scaled boundary coordinate. The infinite element was used to represent the infinite length of the free surface. The initial value of material property(elastic modulus) was considered by the combination of the position of the scaling center and the power function of the radial direction. By use of the mapping type infinite element, the consistent elements formulation could be available. The performance and the feasibility of proposed approach are examined by two numerical examples.

• School Mathematics
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• v.18 no.4
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• pp.819-838
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• 2016

This research analyzed proportional reasoning abilities of the 5th grade students who learned only the basis of ratio and rate and 6th grade students who also learned proportion and cross product strategy. Data were collected through the proportional reasoning tests and the interviews, and then the achievement of the students and their proportional reasoning strategies were analyzed. In the light of such analytical results, the conclusions are as follows. Firstly, there is not much difference between 5th and 6th grade students in the achievement scores. Secondly, both 5th and 6th graders are less familiar with the geometric, qualitative and comparisons tasks than the other tasks. Thirdly, not only 5th graders but also 6th graders used informal strategies much more than the formal strategy. Fourthly, some students can't come up with other strategies than the cross product strategy. Finally, many students have difficulties in discerning proportional situation and non-proportional situations. This study provided suggestions for improving teaching proportional reasoning in elementary schools in Korea as follows: focusing on letting students use their informal strategies fluently in geometric, qualitative, and comparisons tasks as well as algebraic, quantitative, and missing value tasks focusing on the concept of ratio and proportion instead of enforcing the formal strategy.

• Journal of Educational Research in Mathematics
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• v.22 no.3
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• pp.331-352
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• 2012

The purpose of the study is to verify what cognitive variables have significant effect on proportional problem solving. For this aim, the study classified proportional problem into well-structured, moderately-structured, ill-structured problem by the level of structuredness, then classified the cognitive variables as well into factual algorithm knowledge, conceptual knowledge, knowledge of problem type, quantity change recognition and meta-cognition(meta-regulation and meta-knowledge). Then, it verified what cognitive variables have significant effects on 6th graders' proportional problem solving abilities through multiple regression analysis technique. As a result of the analysis, different cognitive variables effect on solving proportional problem classified by the level of structuredness. Through the results, the study suggest how to teach and assess proportional reasoning and problem solving in elementary mathematics class.

• Journal of Elementary Mathematics Education in Korea
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• v.20 no.4
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• pp.601-625
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• 2016

In this study, I hoped to reveal the understanding of pre-service elementary teachers about proportional reasoning and the traits of proportional reasoning strategy used by pre-service elementary teachers. The results of this study are as follows. Pre-service elementary teachers should deal with various proportional reasoning tasks and make a conscious effort to analyze proportional reasoning task and investigate various proportional reasoning strategies through teacher education program. It is necessary that pre-service elementary teachers supplement the lacking tasks such as qualitative reasoning and distinction between proportional situation and non-proportional situation. Finally, It is suggested to preform the future research on teachers' errors and mis-conceptions of proportional reasoning.

• Geotechnical Engineering
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• v.14 no.4
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• pp.103-116
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• 1998

To estimate pure shear strength, 150 sets of triaxial test data were analyzed. The proportional coefficient of shear strength($I_c$) at zero normal stress was nonlinearly decreased as failure coefficient m increases, while the internal friction $\phi_0$ at zero normal stress was nonlinearly increased. The ratio of shear strength $(c/\phi_0)$was inversely proportional to the ratio of the internal friction angles$(\phi/phi_0)$ The shear strength decreased as m increased, while internal friction angle increased. And uniaxial strength was proportional to $c,\phi$ Regression analysis showed that shear strength strongly affects m and $\sigma_c$ The proportional coefficient of shear strength was nonlinearly increased with RMR, while the internal friction angle $(\phi}$was linearly decreased.

• Journal for History of Mathematics
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• v.22 no.3
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• pp.131-152
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• 2009

The Eudoxean theory of Proportion is correlated with 'Dedekind cut' with which Dedekind defined the real number system in modern usage. Dedekind established a firm foundation for the real number system by retracing some of Eudoxus' steps of over two thousand years earlier. Thus it should be quite worthy that we separate Greek inheritance from the definition of Dedekind, However, there is a fundamental difference between Eudoxean theory of proportion and Dedekind cut. Basically, it seems impossible for Greeks to distinguish between the distinction between number and magnitude. In this paper, we will consider how the Eudoxean theory of proportion was related to Dedekind cut introduced to prove the Dedekind's real number completion and how it influenced Dedekind cut by looking at the relation between Eudoxos's explication of the notion of ratio and Dedekind's well-known construction of the real numbers.