• Journal of the Korean School Mathematics Society
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• v.11 no.2
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• pp.285-298
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• 2008

According to the 7th curriculum, irrational numbers should be introduced using infinite decimals in 9th grade. To do so, the relation between rational numbers and decimals should be explained in 8th grade. Preceding studies remarked that middle school students could understand the relation between rational numbers and decimals through the division appropriately. From the point of view with the arithmetic handling activity, I analyzed that the integers and terminating decimals was explained as decimals with repeating 0s or 9s. And, I reviewed the equivalent relations between irrational numbers and non-repeating decimals, rational numbers and repeating decimals. Furthermore, I suggested an alternative method of introducing irrational numbers.

• The Mathematical Education
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• v.63 no.3
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• pp.393-409
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• 2024

The number line model, which intuitively marks numerical magnitudes in space, is widely utilized to help in understanding the magnitudes that fractions and decimals represent. The study analyzed 6^th graders' understanding of fractions and decimals, their problem solving strategies, and whether individual differences in the flexibility of various strategy uses are associated with the accuracy of numerical representation, calculation fluency, and overall mathematical achievement. As a result of the study, students showed relatively lower accuracy in representing fractions and decimals on a number line compared to natural numbers, especially for fractions with odd denominators compared to even denominators, and for two-digit decimals compared to three-digit decimals. Regarding strategy use, students primarily used benchmark, segmentation, and approximation strategies for fractions, and benchmark, rounding, and transformation strategies for decimals sequentially. Lastly, as students used various representation strategies for fractions, their accuracy in representing fractions and their overall mathematical achievement scores showed significantly better outcomes. Taken together, we suggest the need for careful instruction on different interpretations of fractions, the place value of decimals, and the meaning of zero in decimal places. Moreover, we discuss instructional methods that integrate the number line model and its diverse representation strategies to enhance students' understanding of fractions and decimals.

• Journal of Educational Research in Mathematics
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• v.14 no.1
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• pp.1-17
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• 2004

According to the 7-th curriculum, irrational number should be introduced using repeating decimals in 8-th grade mathematics. To do so, the relation between rational numbers and repeating decimals such that a number is rational number if and only if it can be represented by a repeating decimal, should be examined closely Since this relation lacks clarity in some text books, irrational numbers have only slight relation with repeating decimals in those books. Furthermore, some text books introduce irrational numbers showing that $\sqrt{2}$ is not rational number, which is out of 7-th curriculum. On the other hand, if we use numeral 0 as a repetend, many results related to repeating decimals can be represented concisely. In particular, the treatments of order relation with repeating decimals in 8-th grade text books must be reconsidered.

• School Mathematics
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• v.9 no.1
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• pp.99-117
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• 2007

In this paper it is discussed how children develop their logical reasoning beyond difficulties in the process of making sense of division with decimals in the classroom setting. When we consider the gap between mathematics at elementary and secondary levels, and given the logical nature of mathematics at the latter levels, it can be seen as important that the aspects of children's logical development in the upper grades in elementary school should be clarified. This study focuses on the teaching and learning of division with decimals in a 5th grade classroom, because it is well known to be difficult for children to understand the meaning of division with decimals. It is suggested that children begin to conceive division as the relationship between the equivalent expressions at the hypothetical-deductive level detached from the concrete one, and that children's explanation based on a reversibility of reciprocity are effective in overcoming the difficulties related to division with decimals. It enables children to conceive multiplication and division as a system of operations.

• 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.

• Journal of Elementary Mathematics Education in Korea
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• v.15 no.3
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• pp.533-557
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• 2011

The purpose of this study was to understand PCK to improve professionalism of teachers and derive implications about proper teachings methods. For achieving these research purposes, different PCK and teaching methods in class of three teachers(A, B, C) were compared and analyzed targeting division of decimals for 6th grade. For this study, criteria of PCK analysis of teachers was set, PCK questionnaires were produced and distributed, teachers had interviews, PCK of teachers were analyzed, division of decimals class for 6th grade was observed and analyzed, and PCK of teachers and their classes were compared. The implications deriving from comparative analyzing PCK and classes are as follows. First of all, there was a close relation between PCK and classes, leading to a need for efforts of increasing PCK of teachers in every field in order to realize effective classes. Secondly, self study and in-service training are needed to enhance PCK of teachers. Thirdly, more of expertises and materials have to be provided on the instruction manual for teachers.

• Journal of the Korean School Mathematics Society
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• v.10 no.2
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• pp.237-246
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• 2007

According to 7-th curriculum, irrational number should be introduced using non-repeating infinite decimals. A rational number is defined by a number determined by the ratio of some integer p to some non-zero integer q in 7-th grade. In 8-th grade, A number is rational number if and only if it can be expressed as finite decimal or repeating decimal. A irrational number is defined by non-repeating infinite decimal in 9-th grade. There are misconceptions about a non-repeating infinite decimal. Although 1.4532954$\cdots$ is neither a rational number nor a irrational number, many high school students determine 1.4532954$\cdots$ is a irrational number and 0.101001001$\cdots$ is a rational number. The cause of misconceptions is the definition of a irrational number defined by non-repeating infinite decimals. It is a cause of misconception about a irrational number that a irrational number is defined by a non-repeating infinite decimals and the method of using symbol dots in infinite decimal is not defined in text books.

• Journal of Elementary Mathematics Education in Korea
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• v.14 no.2
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• pp.377-399
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• 2010

Mixed calculations involving fractions and decimals covered in the unit 6-Na in elementary school math class cause students difficulties, leading them make lots of errors. If students fail to understand temporarily or partly what the teacher taught or lose confidence and continue to have difficulty due to a lack of understanding and skills of algorithm, though they properly understand the concept and principle of the learning content, it should be resolved through intensive teaching. For students suffering from this problem, a correct diagnosis and appropriate treatment are required. Therefore, this study developed a feedback program after diagnosing students' errors through evaluating them in order to continuously assist them to fully understand contents regarding mixed calculations involving fractions and decimals.

• The Mathematical Education
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• v.50 no.3
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• pp.309-327
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• 2011

In this paper, we extended the division algorithm for the integers to the finite decimals. Though the remainder for the finite decimals is able to be defined as various ways, the remainder could be defined as 'the remained amount' which is the result of the division and as "the remainder" only if 'the remained amount' is decided uniquely by certain conditions. From the definition of "the remainder" for the finite decimal, it could be inferred that 'the division by equal part' and 'the division into equal parts' are proper for the division of the finite decimal concerned with the definition of "the remainder". The finite decimal, based on the unit of measure, seemed to make it possible for us to think "the remainder" both ways: 1" in the division by equal part when the quotient is the discrete amount, and 2" in the division into equal parts when the quotient is not only the discrete amount but also the continuous amount. In this division context, it could be said that the remainder for finite decimal must have the meaning of the justice and the completeness as well. The theorem of the division algorithm for the finite decimal could be accomplished, based on both the unit of measure of "the remainder", and those of the divisor and the dividend. In this paper, the meaning of the division algorithm for the finite decimal was investigated, it is concluded that this theory make it easy to find the remainder in the usual unit as well as in the unusual unit of measure.

• Journal of Elementary Mathematics Education in Korea
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• v.15 no.1
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• pp.1-18
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• 2011

The purpose of this study was to investigate the effects of estimation strategy on placing decimal point in multiplication and division of decimals. To examine the effects of improving calculation ability and reducing decimal point errors with this estimation strategy, the experimental research on operation with decimal was conducted. The operation group conducted the decimal point estimation strategy for operating decimal fractions, whereas the control group used the traditional method with the same test paper. The results obtained in this research are as follows; First, the estimation strategy with understanding a basic meaning of decimals was much more effective in calculation improvement than the algorithm study with repeated calculations. Second, the mathematical problem solving ability - including the whole procedure for solving the mathematical question - had no effects since the decimal point estimation strategy is normally performed after finishing problem solving strategy. Third, the estimation strategy showed positive effects on the calculation ability. Th Memorizing algorithm doesn't last long to the students, but the estimation strategy based on the concept and the position of decimal fraction affects continually to the students. Finally, the estimation strategy assisted the students in understanding the connection of the position of decimal points in the product with that in the multiplicand or the multiplier. Moreover, this strategy suggested to the students that there was relation between the placing decimal point of the quotient and that of the dividend.