• Education of Primary School Mathematics
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• v.27 no.2
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• pp.187-198
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• 2024

The addition and subtraction relationship and the multiplication and division relationship are explicitly dealt with in second and third grade mathematics textbooks. However, these relationships are not discussed anymore in the problem situations and activities in the 4th, 5th, and 6th grade mathematics textbooks. In this study, we investigate the calculation principles of subtraction and division in the elementary school mathematics textbooks. Based on our investigation, we justify the addition and subtraction relationship and the multiplication and division relationship at the level of children's understanding so that we discuss some problem situations and activities where the relationships can be applied to subtraction and division. In addition, we suggest educational implications that can be obtained from children's applying the relationships and the properties of equations to subtraction and division.

• The Mathematical Education
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• v.50 no.3
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• pp.337-354
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• 2011

The purpose of the study was to investigate how students understand multiplication and division of fractions and how their understanding influences the solutions of fractional word problems. Thirteen students from 5th to 6th grades were involved in the study. Students' understanding of operations with fractions was categorized into "a part of the parts", "multiplicative comparison", "equal groups", "area of a rectangular", and "computational procedures of fractional multiplication (e.g., multiply the numerators and denominators separately)" for multiplications, and "sharing", "measuring", "multiplicative inverse", and "computational procedures of fractional division (e.g., multiply by the reciprocal)" for divisions. Most students understood multiplications as a situation of multiplicative comparison, and divisions as a situation of measuring. In addition, some students understood operations of fractions as computational procedures without associating these operations with the particular situations (e.g., equal groups, sharing). Most students tended to solve the word problems based on their semantic structure of these operations. Students with the same understanding of multiplication and division of fractions showed some commonalities during solving word problems. Particularly, some students who understood operations on fractions as computational procedures without assigning meanings could not solve word problems with fractions successfully compared to other students.

• Education of Primary School Mathematics
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• v.26 no.4
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• pp.219-236
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• 2023

This study examined the relationship between preservice teachers' mathematical understanding and problem posing in fractions multiplication and division. To this purpose, 41 preservice teachers performed visual representation and problem posing tasks for fraction multiplication and division, measured their mathematical understanding and problem posing ability, and examined the relationship between mathematical understanding and problem posing ability using cross-tabulation analysis. As a result, most of the preservice teachers showed conceptual understanding of fraction multiplication and division, and five types of difficulties appeared. In problem posing, most of the preservice teachers failed to pose a math problem that could be solved, and four types of difficulties appeared. As a result of cross-tabulation analysis, the degree of mathematical understanding was related to the ability to pose problems. Based on these results, implications for preservice teachers' mathematical understanding and problem posing were suggested.

• The Mathematical Education
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• v.60 no.4
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• pp.409-429
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• 2021

This study investigated instructional methods for fractional division emphasizing algebraic thinking with sixth graders. Specifically, instructional elements for fractional division emphasizing algebraic thinking were derived from literature reviews, and the fractional division instruction was reorganized on the basis of key elements. The instructional elements were as follows: (a) exploring the relationship between a dividend and a divisor; (b) generalizing and representing solution methods; and (c) justifying solution methods. The instruction was analyzed in terms of how the key elements were implemented in the classroom. This paper focused on the fractional division instruction with problem contexts to calculate the quantity of a dividend corresponding to the divisor 1. The students in the study could explore the relationship between the two quantities that make the divisor 1 with different problem contexts: partitive division, determination of a unit rate, and inverse of multiplication. They also could generalize, represent, and justify the solution methods of dividing the dividend by the numerator of the divisor and multiplying it by the denominator. However, some students who did not explore the relationship between the two quantities and used only the algorithm of fraction division had difficulties in generalizing, representing, and justifying the solution methods. This study would provide detailed and substantive understandings in implementing the fractional division instruction emphasizing algebraic thinking and help promote the follow-up studies related to the instruction of fractional operations emphasizing algebraic thinking.

• Journal of Elementary Mathematics Education in Korea
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• v.13 no.1
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• pp.17-30
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• 2009

This paper is about the basic skills of four operations in numbers and operations areas from step 1 to step 3 in elementary mathematics. Here are the results of the evaluation. First, addition and subtraction take the largest time. The average difficulty rate in operations area is 91.2%. Most students understand the contents of textbook well. Specifically, students easily understand the step 1. However, subtraction has lower difficulty rate than addition. Also, three mixed computation, calculation in horizontal, and rounding(rounding down) are difficult areas for students. The contents of step 2 are fully understood. However, lots of mistakes are found in the process of rounding(rounding down), and sentence problems are thought as difficult. Second, the multiplication is first starting in the step 2-Ga. The unit 'Multiplication 99' takes 13 hours, the longest. The difficulty rate in this unit is 89.4%, students understand well. However, students are influenced by addition and subtraction errors in the process of multiplication, and have difficulty in changing the sentence problem to multiplication expression. Third, the division, which starts in step 3-Ga, has 89.9% of difficulty rate. Students well understand. Result of this paper: most of students understand well four operations, but accurate concept, the relationship between multiplication and division, specific instructions in teaching principles of division calculation and sentence problems are in need. Setting the amount of the contents and difficulty rate in understanding are depends on every school's situation, so suggesting universal standard is really hard. However, studying more objects broadly and specific study will be helpful to suggest proper contents and effective teaching.

• The Mathematical Education
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• v.51 no.1
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• pp.21-45
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• 2012

The purpose of the study was to investigate how students understand each operations with whole numbers and fractions, and the relationship between their knowledge of operations with whole numbers and conceptual understanding of operations on fractions. Researchers categorized students' understanding of operations with whole numbers and fractions based on their semantic structure of these operations, and analyzed the relationship between students' understanding of operations with whole numbers and fractions. As the results, some students who understood multiplications with whole numbers as only situations of "equal groups" did not properly conceptualize multiplications of fractions as they interpreted wrongly multiplying two fractions as adding two fractions. On the other hand, some students who understood multiplications with whole numbers as situations of "multiplicative comparison" appropriately conceptualize multiplications of fractions. They naturally constructed knowledge of fractions as they build on their prior knowledge of whole numbers compared to other students. In the case of division, we found that some students who understood divisions with whole numbers as only situations of "sharing" had difficulty in constructing division knowledge of fractions from previous division knowledge of whole numbers.

• 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 Embryo Transfer
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• v.27 no.4
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• pp.237-243
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• 2012

This research was investigated the relationship between the number of the transferable embryos and estrus expression rate, BCS (Body Condition Score), which affect the nutritional state of the cow, in Holstein donor cows. CIDRs were inserted into the vaginas of twenty two head of Holstein cows, regardless of estrous cycle. Superovulation was induced using folliclar stimulating hormone (FSH). For artificial insemination, donor cows were injected with $PGF_{2{\alpha}}$ and estrus was checked about 48 hours after the injection. Then they were treated with 4 straws of semen 3 times, with 12-hour intervals. Embryos were collected by a non-surgical method 7 days after the first artificial insemination. When BCS was $$\leq_-$$2.5, the total number of collected ova was 7.3 + 1.9, which is significantly lower (p<0.05) than the numbers 15.4 + 2.8 and 15.4 + 2.1 that were obtained when BCSs were 2.75 and $$\geq_-$$3.0, respectively. Whereas the numbers of transferable embryos were 5.2 + 1.4 when BCS was $$\leq_-$$2.5, which was smaller than the numbers 6.0 + 2.1 and 8.5 + 1.8 that were obtained when BCSs were 2.75 and $$\geq_-$$3.0, respectively; however, the differences were not significant. As for estrus induction rate, the cow groups whose BCSs were 2.75 and $$\geq_-$$3.0 showed 100.0% and 95.0%, respectively. Whereas the cow group whose BCS was $$\leq_-$$2.5 showed 57.1%, and the differences were significant (p< 0.05). As for estrous expression rate, the cow groups whose BCSs were $$\leq_-$$2.5, 2.75 and $$\geq_-$$3.0 showed 100.0%, 100.0% and 85.7%, respectively; however, the differences were not significant. According to the result of this research, it is considered that the total number of collected ova and the number of transferable embryos will be affected by the nutritional state before and after in vivo embryo production and superovulation treatment, and that although the mechanism is not clear, poor stockbreeding management and nutritional level would cause the decrease of ovum recovery rate and the number of transferable embryos in high-producing cows. On the other hand, diverse researches on the superovulation treatment method that is suitable for high-producing Holstein donor cows would contribute to preventing ovarian cyclicity disorder, as well as to the early multiplication of cows with superior genes by increasing the utilization value of donor cows.

• Structural Engineering and Mechanics
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• v.74 no.1
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• pp.33-54
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• 2020

The aim of this study is to calculate natural frequencies and harmonic responses of cracked frames with general boundary conditions by using transfer matrix method (TMM). The TMM is a straightforward technique to obtain harmonic responses and natural frequencies of frame structures as the method is based on constructing a relationship between state vectors of two ends of structure by a chain multiplication procedure. A single variable shear deformation theory (SVSDT) is applied, as well as, Timoshenko beam theory (TBT) and Euler-Bernoulli beam theory (EBT) for comparison purposes. Firstly, free vibration analysis of intact and cracked frames are performed for different crack ratios using TMM. The crack is modelled by means of a linear rotational spring that divides frame members into segments. The results are verified by experimental data and finite element method (FEM) solutions. The harmonic response curves that represent resonant and anti-resonant frequencies directly are plotted for various crack lengths. It is seen that the TMM can be used effectively for harmonic response analysis of cracked frames as well as natural frequencies calculation. The results imply that the SVSDT is an efficient alternative for investigation of cracked frame vibrations especially with thick frame members. Moreover, EBT results can easily be obtained by ignoring shear deformation related terms from governing equation of motion of SVSDT.

• Journal of Educational Research in Mathematics
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• v.27 no.4
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• pp.833-855
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• 2017

This study aims to reinvestigate the reason for introducing radian as a new unit to express the size of angles, what is the meaning of radian measures to use arc lengths as angle measures, and why is the domain of trigonometric functions expanded to real numbers for expressing general angles. For this purpose, it was conducted historical, mathematical and applied mathematical analyzes in order to research at multidisciplinary analysis of the radian concept. As a result, the following were revealed. First, radian measure is intrinsic essence in angle measure. The radian is itself, and theoretical absolute unit. The radian makes trigonometric functions as real functions. Second, radians should be aware of invariance through covariance of ratios and proportions in concentric circles. The orthogonality between cosine and sine gives a crucial inevitability to the radian. It should be aware that radian is the simplest standards for measuring the length of arcs by the length of radius. It can find the connection with sexadecimal method using the division strategy. Third, I revealed the necessity by distinction between angle and angle measure. It needs justification for omission of radians and multiplication relationship strategy between arc and radius. The didactical suggestions derived by these can reveal the usefulness and value of the radian concept and can contribute to the substantive teaching of radian measure.