• School Mathematics
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• v.7 no.2
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• pp.139-168
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• 2005

The purpose of this study is to investigate children's informal knowledge of the fractional multiplication and to develop a teaching material connecting the informal and the formal knowledge. Six lessons of the pre-teaching material are developed based on literature reviews and administered to the 7 students of the 4th grade in an elementary school. It is shown in these teaching experiments that children's informal knowledge of the fractional multiplication are the direct modeling of using diagram, mathematical thought by informal language, and the representation with operational expression. Further, teaching and learning methods of formalizing children's informal knowledge are obtained as follows. First, the informal knowledge of the repeated sum of the same numbers might be used in (fractional number)$\times$((natural number) and the repeated sum could be expressed simply as in the multiplication of the natural numbers. Second, the semantic meaning of multiplication operator should be understood in (natural number)$\times$((fractional number). Third, the repartitioned units by multiplier have to be recognized as a new units in (unit fractional number)$\times$((unit fractional number). Fourth, the partitioned units should be reconceptualized and the case of disjoint between the denominator in multiplier and the numerator in multiplicand have to be formalized first in (proper fractional number)$\times$(proper fractional number). The above teaching and learning methods are melted in the teaching meterial which is made with corrections and revisions of the pre-teaching meterial.

• School Mathematics
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• v.14 no.2
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• pp.191-212
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• 2012

In the field of arithmetic in mathematics education, there has been lack of fine-grained investigations addressing the relationship between students' construction of division knowledge with fractional quantities and their whole number division knowledge. This study, through the analysis of part of collected data from a year-long teaching experiment, presents a possible constructive itinerary as to how a student could modify her unit-segmenting scheme to deal with various fraction measurement division situations: 1) unit-segmenting scheme with a remainder, 2) fractional unit-segmenting scheme. Thus, this study provides a clue for curing a fragmentary approach to teaching whole number division and fraction division and preventing students' fragmentary understanding of the same arithmetical operation in different number systems.

• Journal of Elementary Mathematics Education in Korea
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• v.12 no.1
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• pp.1-25
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• 2008

The purpose of this study was to identify pre-service teachers' Pedagogical Content Knowledge (PCK) about decimal calculation. A written questionnaire was developed dealing with decimal calculation. A total of 152 pre-service teachers from 3 universities were selected for this study; they had taken an elementary mathematics teaching method course and had no teaching experience. The results were as follows: First, with regard to the method of decimal calculation, most pre-service teachers were familiar with algorithms introduced in the textbook. But with regard to the meaning of decimal calculations, they had difficulties in understanding decimal multiplication or decimal division with decimal number. Second, pre-service teachers recognized reasons of errors as well as errors patterns that student might make. But this recognition was limited mainly to errors related to natural number calculation. Third, pre-service teachers frequently commented about decimals algorithms, picture models, the meanings of decimal calculations, and connections to natural number calculations. Many of them represented the meanings of decimal calculations through picture models as to help students' understanding, while they just mentioned algorithms or treated decimal calculation as natural number calculations with decimal point.

• Education of Primary School Mathematics
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• v.19 no.1
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• pp.19-30
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• 2016

This study examined pre-service teachers' pedagogical content knowledge of fraction division in a context where they were asked to write a story problem for a symbolic expression illustrating a whole number divided by a proper fraction. Problem-posing is an important instructional strategy with the potential to create meaningful contexts for learning mathematical concepts, especially when real-world applications are intended. In this study, story problems written by 135 elementary pre-service teachers were analyzed with respect to mathematical correctness. error types, and division models. Patterns and tendencies in elementary pre-service teachers' knowledge of fraction division were identified. Implicaitons for teaching and teacher education are discussed.

• Journal of Educational Research in Mathematics
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• v.19 no.3
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• pp.461-477
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• 2009

In this study, an operational analysis in the context of linear equations is presented. For the analysis, several second-order models concerning students' whole number knowledge and fraction knowledge based on teaching experiment methodology were employed, in addition to our first-order analysis. This ontogenetic analysis begins with students' Explicitly Nested number Sequence (ENS) and proceeds on through various forms of linear equations. This study shows that even in the same representational forms of linear equations, the mathematical knowledge necessary for solving those equations might be different based on the type of coefficients and constants the equation consists of. Therefore, the pedagogical implications are that teachers should be able to differentiate between different types of linear equation problems and propose them appropriately to students by matching the required mathematical knowledge to the students' potential constructs.

• Journal of Elementary Mathematics Education in Korea
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• v.18 no.1
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• pp.1-17
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• 2014

Instructions for the commutative property of multiplication at elementary schools tend to be based on checking the equality between the quantities of 'a times b 'and b' times a, ' for example, $3{\times}4=12$ and $4{\times}3=12$. This article critically examined the approaches to teach the commutative property of multiplication from Kant's perspective of mathematical knowledge. According to Kant, mathematical knowledge is a priori. Yet, the numeric exploration by checking the equality between the amounts of 'a groups of b' and 'b groups of a' does not reflect the nature of apriority of mathematical knowledge. I suggest we teach the commutative property of multiplication in a way that it helps reveal the operational schema that is necessarily and generally involved in the transformation from the structure of 'a times b' to the structure of 'b times a.' Distributive reasoning is the mental operation that enables children to perform the structural transformation for the commutative property of multiplication by distributing a unit of one quantity across the other quantity. For example, 3 times 4 is transformed into 4 times 3 by distributing each unit of the quantity 3, which results in $3{\times}4=(1+1+1){\times}4=(1{\times}4)+(1{\times}4)+(1{\times}4)+(1{\times}4)=4+4+4=4{\times}3$. It is argued that the distributive reasoning is also critical in learning the subsequent mathematics concepts, such as (a whole number)${\times}10$ or 100 and fraction concept and fraction multiplication.

• Journal of Elementary Mathematics Education in Korea
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• v.14 no.2
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• pp.241-262
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• 2010

This study was carried out to identify the cognitive obstacles while using addition and subtraction with fractions, and to analyze the sources of cognitive obstacles. For this purpose, the following research questions were established : 1. What errors do elementary students make while performing the operations with fractions, and what cognitive obstacles do they have? 2. What sources cause the cognitive obstacles to occur? The results obtained in this study were as follows : First, the student's cognitive obstacles were classified as those operating with same denominators, different denominators, and both. Some common cognitive obstacles that occurred when operating with same denominators and with different denominators were: the students would use division instead of addition and subtraction to solve their problems, when adding fractions, the students would make a natural number as their answer, the students incorporated different solving methods when working with improper fractions, as well as, making errors when reducing fractions. Cognitive obstacles in operating with same denominators were: adding the natural number to the numerator, subtracting the small number from the big number without carrying over, and making errors when doing so. Cognitive obstacles while operating with different denominators were their understanding of how to work with the denominators and numerators, and they made errors when reducing fractions to common denominators. Second, the factors that affected these cognitive obstacles were classified as epistemological factors, psychological factors, and didactical factors. The epistemological factors that affected the cognitive obstacles when using addition and subtraction with fractions were focused on hasty generalizations, intuition, linguistic representation, portions. The psychological factors that affected the cognitive obstacles were focused on instrumental understanding, notion image, obsession with operation of natural numbers, and constraint satisfaction.

• School Mathematics
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• v.6 no.3
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• pp.267-281
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• 2004

The purpose of this study is to understand Preservice elementary teachers' knowledge about multiplication of fractions by focusing on their computation abilities, understanding of meanings, generating appropriate problem contexts and representations. A total of 115 preservice elementary teachers participated in the present study. The results of this study indicated that most of preservice elementary teachers have little difficulty in computing multiplication of fractions for right answers, but they have big difficulty in understanding meanings and generating appropriate problem contexts for multiplication of fractions when the multiplier is not an integer, called 'multiplier effect.' Likewise, the rate of appropriate representations surprisingly decreased for multiplication of fractions when the multiplier is not an integer. The findings also point out that an ability to make problem contexts is highly correlated with representations and meanings. This study implies that teacher education programs need to improve preservice elementary teachers' profound understanding of elementary mathematics in order to fundamentally improve the quality of teaching practices in classrooms.

• The Mathematical Education
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• v.61 no.1
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• pp.1-28
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• 2022

The purpose of this study is to explore how students' fractional knowledge is related to their solving of proportion problems. To this end, 28 clinical interviews with four middle-grade students, each lasting about 30~50 minutes, were carried out from May 2021 to August 2021. The present study focuses on two 7th grade students who exhibited their ability to coordinate three levels of units prior to solving whole number problems. Although the students showed interiorization of three levels of units in solving whole number problems, how they coordinated three levels of units were different in solving proportion problems depending on whether the problems required reasoning with whole numbers or fractions. The students could coordinate three levels of units prior to solving the problems involving whole numbers, they coordinated three levels of units in activity for the problems involving fractions. In particular, the ways the two students employed partitioning operations and how they coordinated quantitative unit structures were different in solving proportion problems involving improper fractions. The study contributes to the field by adding empirical data corroborating the hypotheses that students' ability to transform one three levels of units structure into another one may not only be related to their interiorization of recursive partitioning operations, but it is an important foundation for their construction of splitting operations for composite units.

• 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.