• Communications of Mathematical Education
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• v.19 no.1 s.21
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• pp.1-14
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• 2005

Piaget는 인류가 지식을 구성해 온 방식과 유사한 방식으로 어린이들도 자신들이 학습해야 할 지식을 학습할 수 있다고 가정한다. Kamii는 이 가정을 확인하고자 하는 열망으로 실험교수법을 이용한 수학수업을 실시하였다. 본 고에서는 Kamii가 얻은 결과 중 곱셈에 대한 결과를 발생론적 인식론 입장에서 논의가 이루어 질 것이다. 이 논의는 어린이들이 구성해 가는 지식이 선대인들이 사용하던 지식과 유사하다는 점과 어린이들이 구성해 가는 지식이 완성된 지식의 형태를 갖출 수 있다는 점을 중심으로 이루어진다. 또한, 그 결과로부터 전통적인 수학 교수법에 변화가 있어야 함을 발생론적 인식론을 적용한 수학 수업의 특징과 비교하면서 시사점을 논의하고자 한다.

• 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.

• Communications of Mathematical Education
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• v.31 no.2
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• pp.153-166
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• 2017

164 preservice elementary teachers' decision and enactment of their knowledge of fraction multiplication were examined in a context where they were asked to write a story problem for a multiplication problem with two proper fractions. Participants were selected from an entry level course and an exit level course of their teacher preparation program to reveal any differences between the groups as well as any recognizable patterns within each group and overall. Patterns and tendencies in writing story problems were identified and analyzed. Implications of the findings for teaching and teacher education are discussed.

• Journal of Educational Research in Mathematics
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• v.18 no.4
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• pp.483-497
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• 2008

This study investigated what kind of informal knowledge is emergent and what role informal knowledge play in process of solving 2-digit by 2-digit multiplication task. The data come from 4 times interviews with a 3th grade student who had not yet received regular school education regarding 2-digit by 2-digit multiplication. And the data involves the student's activity paper, the characteristics of action and the clue of thinking process. Findings from these interviews clarify the child's informal knowledge to modeling strategy, doubling strategy, distributive property, associative property. The child formed informal knowledge to justify and modify her conjecture of the algorithm.

• Journal of Elementary Mathematics Education in Korea
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• v.14 no.1
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• pp.135-151
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• 2010

The purpose of this study is to investigate the effect of learner-centered instructions using instructional materials which are developed in the light of constructivism and implementing practices of the instruction. According to the result of Recall Test, experimental group and comparing group have not statistically meaningful difference. However, in the result of Generation Tests which include the contents not dealt with during the experiment treatments, the two groups have statistically meaningful difference. It can be drawn from the result that students who take learner-centered instruction are in a good readiness for learning of the contents which will be addressed in future.

• Journal of Elementary Mathematics Education in Korea
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• v.23 no.1
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• pp.1-17
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• 2019

In this study, prospective teachers were involved in arranging the teaching sequence of multiplication and division of fractions and decimal numbers based on their experience and knowledge of school mathematics. As a result, these activities provided an opportunity to demonstrate the prospective teachers' perception. Prospective teachers were able to learn the knowledge they needed by identifying the differences between their perceptions and curriculum. In other words, prospective teachers were able to understand the mathematical relationships inherent in the teaching sequence of multiplication and division of fractions and decimal numbers and the importance and difficulty of identifying students' prior knowledge and the effects of productive failures as teaching methods.

• 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 the Korean School Mathematics Society
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• v.12 no.1
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• pp.47-70
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• 2009

The purpose of this research was to confirm one of constructivists' assumptions that even children 조o are with low reasoning ability can make reflective abstracting ability and cognitive structures by this ability can make generation ability of new knowledge by themselves. To investigate the assumption, learner-centered instruction were implemented to 2nd grade classroom located in Suseong Gu, DaeGu City and with lesson plans which initially were developed by Burns and corrected by the researchers. Recordings videoed using 2 video cameras, observations, instructions, children's activity worksheets, instruction journals were analyzed using multiple tests for qualitative analysis. Some conclusions are drawn from the results. First, even children with low reasoning ability can construct mathematical knowledge on multiplication in their own. ways, Thus, teachers should not compel them to learn a learning lesson's goals which is demanded in traditional instruction, with having belief they have reasoning ability. Second, teachers need to have the perspectives of respects out of each child in their classroom and provide some materials which can provoke children's cognitive conflict and promote thinking with the recognition of effectiveness of learner-centered instruction. Third, students try to develop their ability of reflective and therefore establish cognitive structures such as webs, not isolated and fragmental ones.

• Journal of Educational Research in Mathematics
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• v.16 no.2
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• pp.157-177
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• 2006

The concepts of ratio and proportion do not develop in isolation. Rather, they are part of the individual's multiplicative conceptual field, which includes other concepts such as multiplication, division, and rational numbers. The current study attempted to clarify the beginning of this development process. One fourth student, Kyungsu, was encourage to schematize his trial-and-error-based method, which was effective in solving so-called missing-value tasks. This study describes several advancements Kyungsu made during the teaching experiment and analyzes the challenges Kyungsu faced in attempting to schematize his method. Finally, the mathematical knowledge Kyungsu needed to further develop his ratio and proportion concepts is identified. The findings provide additional support for the view that the development of ratio and proportion concepts is embedded within the development of the multiplicative conceptual field.

• 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.