• Journal of Educational Research in Mathematics
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• v.27 no.4
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• pp.765-789
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• 2017

Arithmetic is the basis of school mathematics and in fact, number and operation in elementary school curriculum is the most basic and essential domain. Even though there has been a consensus that arithmetic should be taught more meaningfully beyond the emphasis of calculation skills and teachers should emphasize the aspect of the arithmetical thinking, it is difficult to find studies which focus on the arithmetical thinking itself. So this research aims to explore the meaning of the arithmetical thinking and extract the arithmetical thinking factors. In order to solve the research problems, we reviewed and analyzed the literatures and then conducted Delphi survey to extract arithmetical thinking factors. From the results of this research, we found the meaning of arithmetical thinking and the arithmetical thinking factors. Especially, the arithmetical thinking consists of 18 factors. It is important to pay attention to students' arithmetical thinking because there are various factors of the arithmetical thinking. It is necessary to identify the aspects of arithmetical thinking reflected in school mathematics based on the meaning of arithmetical thinking and its factors. Based on this, it is possible to find effective teaching and learning methods of arithmetic focusing on the arithmetical thinking.

• Journal of Elementary Mathematics Education in Korea
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• v.24 no.1
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• pp.89-108
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• 2020

Since arithmetic is the foundation of school mathematics, it needs to be taught meaningfully in the direction of improving arithmetical thinking levels of students beyond the fluency of computing skills. Therefore, in this study, the arithmetical thinking levels of 100 students in 5th grade were analyzed by applying the arithmetical thinking level test. As a result, 82 students were at 1st level and 15 students were at 2nd level of the arithmetical thinking. I analyzed the characteristics of arithmetical thinking and types of errors and misconceptions made by the students, and derived some didactical implications for arithmetic education in elementary school mathematics.

• Journal of Elementary Mathematics Education in Korea
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• v.21 no.4
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• pp.575-598
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• 2017

This study aims to explore the level-specific characteristics of arithmetical thinking based on the arithmetical thinking factors and develop an arithmetical thinking level test that can identify students' arithmetical thinking levels by specifying the levels of arithmetical thinking based on the factors. In order to solve the research problems, we categorized the arithmetical thinking factors into 1~4 levels based on the literature review and constructed items of the arithmetical thinking level test considering both content and process based on the arithmetical thinking factors and the level-specific characteristics of the arithmetical thinking which conformed to the Guttman scale. To investigate the adequacy of the analysis of the arithmetical thinking levels, we reanalyzed the level-specific characteristics of the arithmetical thinking by checking that it matched the factors classified to the test developed by the Guttman scale. From the results of this research, the following conclusions were drawn. First, the arithmetical thinking factors are categorized into four levels which have different characteristics. Second, the arithmetical thinking level test of this study was developed satisfying the Guttman scale and it reflects the level-specific characteristics of the arithmetical thinking levels from 1 to 4. It is possible to determine the students' arithmetical thinking level using this test. Third, according to the results of the final application of the arithmetical thinking level test for 5th and 6th graders, teachers should provide more abundant learning experiences related to the relation level (the level 3) and the application level (the level 4) to increase students' arithmetical thinking level.

• Journal of Educational Research in Mathematics
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• v.26 no.3
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• pp.489-508
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• 2016

Number and operation is the most basic and crucial part in elementary mathematics but is also well known as a part that students have lots of difficulties. A lot of researches have been done in various ways to solve this problem but it can't be solved fundamentally by emphasizing calculation method and skill. So we need to go over it in terms of relevant arithmetical thinking. This study aims to diagnose the cause of an underachiever's difficulties about arithmetic and finds a prescription for her by analyzing her level of arithmetical thinking based on Guberman(2014) and understanding about arithmetic. To achieve this goal, we chose an 6th grader who's having a hard time particularly in number and operation among mathematics strands and conducted a case study carrying out arithmetical thinking level tests on two separate occasions and analyzing her responses. As a result of analyzing data, her arithmetical thinking corresponded to Guberman's first level and it is also turned out that student is suffering from some arithmetic concepts. We suggest several implications for teaching of arithmetic at elementary school in terms of the development of arithmetical thinking based on analysis result and discussion about it.

• Communications of Mathematical Education
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• v.22 no.2
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• pp.137-164
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• 2008

Given the importance of early experience in algebraic thinking, we designed six consecutive lessons in which $4^{th}$ graders were encouraged to recognize patterns in the process of finding the relationships between two quantities and to represent a given problem with various mathematical models. The results showed that students were able to recognize patterns through concrete activities with manipulative materials and employ various mathematical models to represent a given problem situation. While students were able to represent a problem situation with algebraic expressions, they had difficulties in using the equal sign and letters for the unknown value while they attempted to generalize a pattern. This paper concludes with some implications on how to connect algebraic thinking with students' arithmetic or informal thinking in a meaningful way, and how to approach algebra at the elementary school level.

• Journal of Educational Research in Mathematics
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• v.17 no.4
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• pp.453-475
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• 2007

School algebra starts with introducing algebraic expressions which have been one of the cognitive obstacles to the students in the transfer from arithmetic to algebra. In the recent studies on the teaching school algebra, algebraic thinking is getting much more attention together with algebraic expressions. In this paper, we examined the processes of the transfer from arithmetic to algebra and ways for teaching early algebra through algebraic thinking factors. Issues about algebraic thinking have continued since 1980's. But the theoretic foundations for algebraic thinking have not been founded in the previous studies. In this paper, we analyzed the algebraic thinking in school algebra from historico-genetic, epistemological, and symbolic-linguistic points of view, and identified algebraic thinking factors, i.e. the principle of permanence of formal laws, the concept of variable, quantitative reasoning, algebraic interpretation - constructing algebraic expressions, trans formational reasoning - changing algebraic expressions, operational senses - operating algebraic expressions, substitution, etc. We also identified these algebraic thinking factors through analyzing mathematics textbooks of elementary and middle school, and showed the middle school students' low achievement relating to these factors through the algebraic thinking ability test. Based upon these analyses, we argued that the readiness for algebra learning should be made through the processes including algebraic thinking factors in the elementary school and that the transfer from arithmetic to algebra should be accomplished naturally through the pre-algebra course. And we searched for alternative ways to improve algebra curriculums, emphasizing algebraic thinking factors. In summary, we identified the problems of school algebra relating to the transfer from arithmetic to algebra with the problem of teaching algebraic thinking and analyzed the algebraic thinking factors of school algebra, and searched for alternative ways for improving the transfer from arithmetic to algebra and the teaching of early algebra.

• Education of Primary School Mathematics
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• v.14 no.3
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• pp.261-275
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• 2011

We observed the process for solving linear equations of two 5th grade elementary students, who do not have any pre-knowledge about solving linear equation. The way of students' usage of fractional schemes and manipulations are closely observed. The change of their scheme adaptation are carefully analyzed while the coefficients and constants become complicated. The results showed that they used various fractional scheme and manipulations according to the coefficients and constants. Noticeably, they used repeating fractional schemes to establish the equivalence relation between unknowns and the given quantities. After establishing the relationship, equivalent fractions played important role. We expect the results of this study would help shorten the gap between the arithmetic and the algebraic thinking.

• The Mathematical Education
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• v.42 no.5
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• pp.697-706
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• 2003

In this paper we consider the equality sign as a mathematical concept and investigate its meaning, errors made by students, and subject matter knowledge of mathematics teacher in view of The Model of Mathematic al Concept Analysis, arithmetic-algebraic thinking, and some examples. The equality sign = is a symbol most frequently used in school mathematics. But its meanings vary accor ding to situations where it is used, say, objects placed on both sides, and involve not only ordinary meanings but also mathematical ideas. The Model of Mathematical Concept Analysis in school mathematics consists of Ordinary meaning, Mathematical idea, Representation, and their relationships. To understand a mathematical concept means to understand its ordinary meanings, mathematical ideas immanent in it, its various representations, and their relationships. Like other concepts in school mathematics, the equality sign should be also understood and analysed in vie w of a mathematical concept.

• Journal of Educational Research in Mathematics
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• v.27 no.4
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• pp.727-745
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• 2017

This study investigated the Aristotle's continuity and the historical development of continuity of function to explore the differences between the concepts of mathematics and students' thinking about continuity of functions. Aristotle, who sought the essence of continuity, characterized continuity as an 'indivisible unit as a whole.' Before the nineteenth century, mathematicians considered the continuity of functions based on space, and after the arithmetization of nineteenth century modern ${\epsilon}-{\delta}$ definition appeared. Some scholars thought the process was revolutionary. Students tended to think of the continuity of functions similar to that of Aristotle and mathematicians before the arithmetization, and it is inappropriate to regard students' conceptions simply as errors. This study on the continuity of functions examined that some conceptions which have been perceived as misconceptions of students could be viewed as paradigmatic thoughts rather than as errors.

• Journal of Korean Society of Transportation
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• v.22 no.1 s.72
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• pp.63-72
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• 2004

In conducting traffic safety programs, it is very important to identify hazardous sites in appropriate manner. The rate qualify control method is generally used in identifying hazardous sites since it can interpret the sites in the statistic aspects. The rate qualify control method is based on the assumption that the occurrences of traffic crashes follow the Poisson's distribution in which the expected value of traffic crashes equals the variance of those. However, there is greater variability than expected statistically, we call this phenomenon over dispersion. This study analyzed the problem related to the rate quality control method under the over dispersed data, and established a methodology to solve the problem. As a result of test on the basis of the field data, the new approach produced more reasonable results than those of the Poisson based rate quality control method.