• Journal of Elementary Mathematics Education in Korea
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• v.13 no.2
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• pp.211-229
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• 2009

The purpose of this study is analysis of to investigate relation proportion problem of mathematics textbooks of 7th curriculum to proportional reasoning(relative thinking, unitizing, partitioning, ratio sense, quantitative and change, rational number) of Lamon's proposal at sixth grade students. For this study, I develop two test papers; one is for proportion problem of mathematics textbooks test paper and the other is for proportional reasoning test paper which is devided in 6 by Lamon. I test it with 2 group of sixth graders who lived in different region. After that I analysis their correlation. The result of this study is following. At proportion problem of mathematics textbooks test, the mean score is 68.7 point and the score of this test is lower than that of another regular tests. The percentage of correct answers is high if the problem can be solved by proportional expression and the expression is in constant proportion. But the percentage of correct answers is low, if it is hard to student to know that the problem can be expressed with proportional expression and the expression is not in constant proportion. At proportion reasoning test, the highest percentage of correct answers is 73.7% at ratio sense province and the lowest percentage of that is 16.2% at quantitative and change province between 6 province. The Pearson correlation analysis shows that proportion problem of mathematics textbooks test and proportion reasoning test has correlation in 5% significance level between them. It means that if a student can solve more proportion problem of mathematics textbooks then he can solve more proportional reasoning problem, and he have same ability in reverse order. In detail, the problem solving ability level difference between students are small if they met similar problem in mathematics text book, and if they didn't met similar problem before then the differences are getting bigger.

• 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.

• School Mathematics
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• v.15 no.4
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• pp.723-742
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• 2013

The purpose of this study was to analyze children's proportional reasoning process on an ill-structured "architectural drawing" problem solving and to investigate their level and characteristics of proportional reasoning. As results, they showed various perspective and several level of proportional reasoning such as illogical, additive, multiplicative, and functional approach. Furthermore, they showed their expanded proportional reasoning from the early stage of perception of various types of quantities and their proportional relation in the problem to application stage of their expanded and generalized relation. Students should be encouraged to develop proportional reasoning by experiencing various quantity in ration and proportion situations.

• Education of Primary School Mathematics
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• v.12 no.2
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• pp.117-132
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• 2009

The purpose of this study was analysis on types of strategies and errors when the sixth grade students were solving proportion problems of mathematics textbooks. For this study, proportion problems in mathematics textbooks were investigated and 17 representative problems were chosen. The 277 students of two elementary schools solved the problems. The types of strategies and errors in solving proportion problems were analyzed. The result of this study were as follows; The percentage of correct answers is high if the problems could be solved by proportional expression and the expression is in constant rate. But the percentage of correct answers is low, if the problems were expressed with non-constant rate.

• Journal of The Korean Association For Science Education
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• v.24 no.5
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• pp.987-995
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• 2004

The purpose of this study was to analyze characteristics of problem solving process with proportional or compensational reasoning of the elementary school students. For this study, 85th grade students were selected and tested with Science Reasoning Task, information processing ability test and proportional and compensational reasoning tasks. This study revealed that students in mid concrete stage could solve the proportionality task and easy compensation task. But, most of the students could not solve difficult compensation task. And as the students got higher score in information processing test, it took them less time to solve the problem. The types of strategy used in solving proportional and compensational problem were categorized as the factor of change, building-up and the cross-product. Most of the students failed in problem solving used incorrect schema knowledge, procedure knowledge and strategy knowledge. Many students tended to use proportionality strategy to solve the difficult compensation task. Result of this study suggested that various task included different structure and the same schema knowledge can be effective for the advancement of students' proportional and compensational reasoning ability.

• 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.

• 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.

• Education of Primary School Mathematics
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• v.15 no.2
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• pp.59-75
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• 2012

This study aims to find whether the solutions for well-structured problems learned in school can be transferred to the moderately-structured problem and ill-structured problem. For these purpose, research questions were set up as follows: First, what are the patterns of changes in strategies used in solving the mathematics problems with different level of structuredness? Second, From the group using and not using proportion algorithm strategy in solving moderately-structured problem and ill-structured problem, what features were observed when they were solving that problems? Followings are the findings from this study. First, for the lower level of structuredness, the frequency of using multiplicative strategy was increased and frequency of proportion algorithm strategy use was decreased. Second, the students who used multiplicative strategies and proportion algorithm strategies to solve structured and ill-structured problems exhibited qualitative differences in the degree of understanding concept of ratio and proportion. This study has an important meaning in that it provided new direction for transfer and analogical problem solving study in mathematics education.

• Journal of Educational Research in Mathematics
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• v.23 no.1
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• pp.1-16
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• 2013

The primary purpose of this study is to survey multiplicative thinking levels and its characteristics of third graders in elementary school and to analyze how to use it when they solve the proportional problems. As results, the transition thinking ranked the highest among the four kinds of thinking levels when the $3^{rd}$ graders solved the multiplication problems. It means that the largest numbers of students still can not distinguish the additive and multiplicative situations completely and remain in the transition thinking, which thinks both additively and multiplicatively. In addition, the performance of solving proportional problems was distinguished from the levels of thinking. Through this study, we can give some implications of the importance of multiplicative thinking and instructional methods related to multiplication.

• The Mathematical Education
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• v.61 no.3
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• pp.441-456
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• 2022

The purpose of the current study is to report a part of our larger project whose focus is to understand a relationship between students' units coordination and K-12 school mathematics. In particular, in this paper we report how students who exhibit distinct levels of units coordinations used their knowledge of proportion to solve quadratic function problems of the form y = ax². To this end, three 7th grade students all of whom assimiliated whole number problem situations with three levels of units but showed different levels for fraction problems were chosen. We carried out clinical interviews not only to understand their ability to coordinate units but to understand their problem solving process of proportion and the quadratic function problems. The analysis suggest that their abilities to coordinate units influenced their ways to solving proportion problems, and in turn influenced their ways to solve the specific form of quadratic functions. We have finalized our study by discussing how students' ability to construct and coordinate units, their proportion knowledge, and their knowledge associated with expressing the specific type of quadractic functions could be related.