• Education of Primary School Mathematics
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• v.22 no.3
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• pp.181-203
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• 2019

Even though the properties of operations in multiplication serve a fundamental basis of conceptual understanding the multiplication with whole numbers for elementary students, there has been lack of research in this field. Given this, the purpose of this study was to analyze instructional methods related to the properties of operations in multiplication (i.e., commutative property of multiplication, associative property of multiplication, distributive property of multiplication over addition) in a series of mathematics textbooks of Korea, Japan, and the US. The overall analysis was conducted in the following two aspects: (a) when and how to deal with the properties of multiplication in three instructional context (i.e., introduction, application, generalization), and (b) what models use to represent the properties of multiplication. The results of this showed that overall similarities in introducing the properties of multiplication .in (one digit) ${\times}$ (one digit) as well as emphasizing the divers representation. However, subtle but meaningful differences were analyzed in applying and generalizing the properties of multiplication. Based on these results, this paper closes with some implications on how to teach the properties of operations in multiplication properties in elementary mathematics.

• Journal of Educational Research in Mathematics
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• v.21 no.3
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• pp.239-259
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• 2011

This study investigated the elementary school students' ability on the algebraic reasoning as generalized arithmetic. It analyzed the written responses from 648 second graders, 688 fourth graders, and 751 sixth graders using tests probing their understanding of the properties of whole number operations. The result of this study showed that many students did not recognize the properties of operations in the problem situations, and had difficulties in applying such properties to solve the problems. Even lower graders were quite successful in using the commutative law both in addition and subtraction. However they had difficulties in using the associative and the distributive law. These difficulties remained even for upper graders. As for the associative and the distributive law, students had more difficulties in solving the problems dealing with specific numbers than those of arbitrary numbers. Given these results, this paper includes issues and implications on how to foster early algebraic reasoning ability in the elementary school.

• Education of Primary School Mathematics
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• v.8 no.1
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• pp.1-11
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• 2004

Whole number concept is an important part of mathematics learning. Elementary school students should understand and master whole number concept. Models can help students learn whole number concepts. Realistic Mathematics Education in Netherlands had investigated and developed three different structural models: line model, group model, combination model. These models are related whole number concepts and structures closely and compatible. The purpose of this study is to investigate models related with whole numbers in 7th elementary textbooks. As a result of analyses, we found out some pattern and several problems. On the foundation of the analyses, we discussed the better ways of using models in teaching whole number and the research issues concerning these.

• The Mathematical Education
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• v.50 no.1
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• pp.41-59
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• 2011

Given the importance of algebra in the early grades, this paper analyzed the contents of whole numbers and their operations from the perspectives of generalized arithmetic. In particular, the focus of analysis was given to the properties of 0 and 1, those of operations such as commutativity, associativity, and distributivity, and the relations between operations. As such, this paper analyzed in detail how such properties and relations were introduced and expanded across different grades. It is expected that many issues in this paper will serve basic information to develop instructional materials in a way to fostering students' algebraic thinking in the elementary grades.

• 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.

• Journal of Educational Research in Mathematics
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• v.17 no.2
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• pp.147-162
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• 2007

This paper reports an analysis of 19 Chinese and Korean middles school mathematics teachers' understanding of division by fractions. The study analyzes the teachers' responses to the teaching task of generating a real-world situation representing the meaning of division by fractions. The findings of this study suggests that the teachers' conceptual models of division are dominated by the partitive model of division with whole numbers as equal sharing. The dominance of partitive model of division constraints the teachers' ability to generate real-world representations of the meaning of division by fractions, such that they are able to teach only the rule-based algorithm (invert-and-multiply) for handling division by fractions.

• Journal of Educational Research in Mathematics
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• v.25 no.4
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• pp.657-673
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• 2015

In this study, we are interested in the teachers' MCK about '$N{\div}0$' and MPCK in relation to the proper ways to teach it. Even though '$N{\div}0$' is not on the current curriculum and textbooks of elementary school mathematics, a few students sometimes ask a question about it because the division of the form '$a{\div}b$' is dealt in whole number including 0. Teacher's obvious understanding and appropriate guidance based on students' levels can avoid students' error and have positive effects on their subsequent learning. Therefore, we developed an interview form to investigate teachers' MCK about '$N{\div}0$' and MPCK of the proper ways to teach it and carried out individual interviews with 30 elementary school teachers. The results of the analysis of these interviews reveal that some teachers do not have proper MCK about '$N{\div}0$' and many of them have no idea on how to teach their students who are asking about '$N{\div}0$'. Based on our discussion of the results, we suggest some didactical implications.