본 연구는 아이들이 문장제 또는 수식 형태의 나눗셈의 결과를 여러 타입의 분수들-진분수, 가분수, 대분수-과 연관시키면서 분수가 가지는 여러 하위 개념 중 몫에 대한 개념 도식을 어떻게 구성해 가는지에 대하여 미국의 5학년 초등학생 네 명을 대상으로 이루어졌다. 실험 결과는 다음과 같았다. 균등분배 상황에서, 아이들은 나눗셈을 두 가지 방식으로 개념화하였다. 첫째, 아이들이 나눗셈을 통해 대분수 형태의 몫을 산출했을 경우, 이 대분수 형태의 몫은 진분수와 가분수 형태의 분수들을 부분-전체의 하위개념이 아니라 몫이라는 하위개념으로 이해하는데 개념적인 기초가 되었다. 둘째, 진분수 형태의 몫을 얻은 경우, 아이들은 그 몫을 곱셈구조의 예로 보려는 경향이 있었다. 즉, $a{\times}b=c$ ; $a{\div}c=\frac{1}{b}$ ; $b{\div}c=\frac{1}{a}$. 하지만, 장제법 계산은 소수 형태의 몫을 생산함으로써 아이들이 이 구조를 깨닫는 것을 어렵게 했다.
본 연구는 아이들이 문장제 또는 수식 형태의 나눗셈의 결과를 여러 타입의 분수들-진분수, 가분수, 대분수-과 연관시키면서 분수가 가지는 여러 하위 개념 중 몫에 대한 개념 도식을 어떻게 구성해 가는지에 대하여 미국의 5학년 초등학생 네 명을 대상으로 이루어졌다. 실험 결과는 다음과 같았다. 균등분배 상황에서, 아이들은 나눗셈을 두 가지 방식으로 개념화하였다. 첫째, 아이들이 나눗셈을 통해 대분수 형태의 몫을 산출했을 경우, 이 대분수 형태의 몫은 진분수와 가분수 형태의 분수들을 부분-전체의 하위개념이 아니라 몫이라는 하위개념으로 이해하는데 개념적인 기초가 되었다. 둘째, 진분수 형태의 몫을 얻은 경우, 아이들은 그 몫을 곱셈구조의 예로 보려는 경향이 있었다. 즉, $a{\times}b=c$ ; $a{\div}c=\frac{1}{b}$ ; $b{\div}c=\frac{1}{a}$. 하지만, 장제법 계산은 소수 형태의 몫을 생산함으로써 아이들이 이 구조를 깨닫는 것을 어렵게 했다.
This paper investigated the conceptual schemes four children constructed as they related division number sentences to various types of fraction: Proper fractions, improper fractions, and mixed numbers in both contextual and abstract symbolic forms. Methods followed those of the constructivist teachi...
This paper investigated the conceptual schemes four children constructed as they related division number sentences to various types of fraction: Proper fractions, improper fractions, and mixed numbers in both contextual and abstract symbolic forms. Methods followed those of the constructivist teaching experiment. Four fifth-grade students from an inner city school in the southwest United States were interviewed eight times: Pre-test clinical interview, six teaching / semi-structured interviews, and a final post-test clinical interview. Results showed that for equal sharing situations, children conceptualized division in two ways: For mixed numbers, division generated a whole number portion of quotient and a fractional portion of quotient. This provided the conceptual basis to see improper fractions as quotients. For proper fractions, they tended to see the quotient as an instance of the multiplicative structure: $a{\times}b=c$ ; $a{\div}c=\frac{1}{b}$ ; $b{\div}c=\frac{1}{a}$. Results suggest that first, facility in recall of multiplication and division fact families and understanding the multiplicative structure must be emphasized before learning fraction division. Second, to facilitate understanding of the multiplicative structure children must be fluent in representing division in the form of number sentences for equal sharing word problems. If not, their reliance on long division hampers their use of syntax and their understanding of divisor and dividend and their relation to the concepts of numerator and denominator.
This paper investigated the conceptual schemes four children constructed as they related division number sentences to various types of fraction: Proper fractions, improper fractions, and mixed numbers in both contextual and abstract symbolic forms. Methods followed those of the constructivist teaching experiment. Four fifth-grade students from an inner city school in the southwest United States were interviewed eight times: Pre-test clinical interview, six teaching / semi-structured interviews, and a final post-test clinical interview. Results showed that for equal sharing situations, children conceptualized division in two ways: For mixed numbers, division generated a whole number portion of quotient and a fractional portion of quotient. This provided the conceptual basis to see improper fractions as quotients. For proper fractions, they tended to see the quotient as an instance of the multiplicative structure: $a{\times}b=c$ ; $a{\div}c=\frac{1}{b}$ ; $b{\div}c=\frac{1}{a}$. Results suggest that first, facility in recall of multiplication and division fact families and understanding the multiplicative structure must be emphasized before learning fraction division. Second, to facilitate understanding of the multiplicative structure children must be fluent in representing division in the form of number sentences for equal sharing word problems. If not, their reliance on long division hampers their use of syntax and their understanding of divisor and dividend and their relation to the concepts of numerator and denominator.
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제안 방법
At this time, the effect of contemplating equivalent fractions was the major focus of a study. After that, to reduce the biased result caused by introducing fraction forms in different order, and to maximize the effect of the order of introducing fraction forms, the researcher divided children into different orders of presentation. In addition, the two levels of children’s prior mathematical performance were considered as much as possible when the researcher distributed the children into two presentation orders.
The students conceptualized improper fractions as equal sharing at first. After that, two students conceptualized mixed fractions as equal sharing situations by converting them to improper fractions. For proper fractions, no one conceptualized them to equal sharing situations voluntarily because their part-whole relationship was too strong.
Data analysis consisted of an ongoing analysis phase and a retrospective phase (Huberman & Miles, 1994).
During the six teaching episodes, the students first conceptualized equal sharing word problems where the number of shared objects was bigger than the number of sharers as mixed fractions, and then they conceptualized equal sharing word problems in which the number of shared objects was smaller than the number of sharers as proper fractions. The reason that conceptualizing equal sharing as proper fraction came later than mixed numbers was the problems children had quantifying the result of their partitioning.
For this study, two clinical interviews were conducted with each student to assess his/her initial mathematical knowledge under investigation and to aim at the changes in their mathematical thinking each. Identical problem sets were used for both interviews.
It is based on Toluk's work(1999) and is revised by the researcher based on pilot tests for a clinical interview.
Six teaching episodes followed the initial clinical interview used to construct models of the children’s mathematical thinking and guide them to develop more reflective ways of thinking about partitive quotient fraction that built on the results from the initial clinical interview.
Six teaching episodes followed the initial clinical interview used to construct models of the children’s mathematical thinking and guide them to develop more reflective ways of thinking about partitive quotient fraction that built on the results from the initial clinical interview. The analysis phases involved examining the data from the clinical interviews and the teaching episodes. The purpose of the analysis was to test research the hypotheses and to generate and test ad hoc hypotheses during the teaching episodes by planning, testing, and revising following teaching episodes (Steffe & Thompson, 2000).
The purpose of the analysis was to test research the hypotheses and to generate and test ad hoc hypotheses during the teaching episodes by planning, testing, and revising following teaching episodes (Steffe & Thompson, 2000).
The researcher used probing questions(e.g., “Show me what you did”; “Why did you do that?”; “Show me how that works”) in an attempt to draw out verbal, gestural, and written evidence of their thinking.
The purpose of the analysis was to test research the hypotheses and to generate and test ad hoc hypotheses during the teaching episodes by planning, testing, and revising following teaching episodes (Steffe & Thompson, 2000). Therefore, both ongoing and retrospective analyses were conducted.
This paper investigated how children developed an abstract conception of quotient, and how they struggled overcoming the difficulty that their partitioning strategies hindered the direct mapping between fraction number representations and division representations of quotient situations. Therefore, the major research question is: What conceptual schemes do children construct as they relate division sentences to various fraction forms?
대상 데이터
Four students from a fifth grade mathematics class in an inner city school in the southwest United States participated in the study. One boy and one girl were chosen from the high mathematics performance group and the other boy and the other girl were chosen from the low mathematics performance group each based on their mathematics teacher nomination (see [Table III-1]).
성능/효과
First, they did not consider the situations separately if number of shared objects and the number of sharers had a common factor or not. Second, they did not separate situations where the number of shared objects was bigger than the number of sharers where children preserved the whole number portion of quotient as mixed numbers, from situations where the number of shared objects was bigger than the number of sharers but where the shared objects could only be grouped if conceptualized at a higher level categorization (e.g. red markers and blue markers are both markers), so children dealt with each shared object and got an improper fraction answer.
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