The Mathematical Education (한국수학교육학회지시리즈A:수학교육)

Korean Society of Mathematical Education (한국수학교육학회)
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An analysis of solution methods by fifth grade students about 'reverse fraction problems'

'역 분수 문제'에 대한 5학년 학생들의 해결 방법 분석

  • Pang, JeongSuk (Department of Elementary Education (Mathematics Education), Korea National University of Education) ;
  • Cho, SeonMi (Graduate School of Korea National University of Education)
  • Received : 2018.10.18
  • Accepted : 2018.12.20
  • Published : 2019.02.28
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Abstract

As the importance of algebraic thinking in elementary school has been emphasized, the links between fraction knowledge and algebraic thinking have been highlighted. In this study, we analyzed the solution methods and characteristics of thinking by fifth graders who have not yet learned fraction division when they solved 'reverse fraction problems' (Pearn & Stephens, 2018). In doing so, the contexts of problems were extended from the prior study to include the following cases: (a) the partial quantity with a natural number is discrete or continuous; (b) the partial quantity is a natural number or a fraction; (c) the equivalent fraction of partial quantity is a proper fraction or an improper fraction; and (d) the diagram is presented or not. The analytic framework was elaborated to look closely at students' solution methods according to the different contexts of problems. The most prevalent method students used was a multiplicative method by which students divided the partial quantity by the numerator of the given fraction and then multiplied it by the denominator. Some students were able to use a multiplicative method regardless of the given problem contexts. The results of this study showed that students were able to understand equivalence, transform using equivalence, and use generalizable methods. This study is expected to highlight the close connection between fraction and algebraic thinking, and to suggest implications for developing algebraic thinking when to deal with fraction operations.

Keywords

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[그림 1] ‘역 분수 문제’에 대한 학생들의 해결 방법의 예 (Pearn & Stephens, 2018, p. 243) [Fig. 1] An example of students’ solution methods about Reverse Fraction Problems

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[그림 2] ‘역 분수 문제’와 관련된 우리나라 교과서 문제의 예(교육부, 2018, p. 114) [Fig. 2] An example of problems in the mathematics textbooks related to Reverse Fraction Problems

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[그림 3] 곱셈 방법의 예 [Fig. 3] An example of multiplicative methods

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[그림 4] 곱셈 방법의 다른 예 [Fig. 4] Another example of multiplicative methods

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[그림 5] 부분적 곱셈 방법의 예 [Fig. 5] An example of partially multiplicative methods

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[그림 6] 덧셈 방법의 예 [Fig. 6] An example of addictive methods

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[그림 7] 그림 방법의 예 ① [Fig. 7] The first example of visual methods

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[그림 8] 그림 방법의 예 ② [Fig. 8] The second example of visual methods

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[그림 9] 그림 방법의 예 ③ [Fig. 9] The third example of visual methods

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[그림 10] 그림 방법의 예 ④ [Fig. 10] The fourth example of visual methods

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[그림 11] 부분에 해당하는 양이 이산량인지 연속량인지에 따른 정답률 [Fig. 11] A percentage of correct answers according to the problem contexts in which the partial quantity with a natural number is discrete or continuous

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[그림 12] 부분에 해당하는 양이 자연수인지 분수인지에 따른 정답률 [Fig. 12] A percentage of correct answers according to the problem contexts in which the partial quantity is a natural number or a fraction

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[그림 13] 부분에 해당하는 양이 분수인 맥락에서 곱셈 방법을 사용하여 문제를 해결한 예 [Fig. 13] An example of multiplicative methods for the problems in which the partial quantity with a fraction is continuous

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[그림 14] 부분에 해당하는 양을 나타내는 분수가 진분수인지 가분수인지에 따른 정답률 [Fig. 14] A percentage of correct answers according to the problem contexts in which the equivalent fraction of partial quantity is a proper fraction or an improper fraction

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[그림 15] 그림 제시 여부에 따른 정답률 [Fig. 15] A percentage of correct answers according to the problem contexts in which the diagram is presented or not

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[그림 16] 형식화 방법의 예 [Fig 16] An example of advanced multiplicative methods

[표 1] Pearn과 Stephens(2018)에서 제시한 ‘역 분수 문제’ [Table 1] Reverse Fraction Problems presented in Pearn and Stephens (2018)

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[표 2] 본 연구에서 사용한 ‘역 분수 문제’ [Table 2] Reverse Fraction Problems presented in this study

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[표 3] ‘역 분수 문제’에 대한 학생들의 해결 방법 설명 및 예 [Table 3] Explanations and examples of students’ solution methods used for Reverse Fraction Problems

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[표 4] 학생들이 사용한 문제 해결 방법 (N=46) [Table 4] Solution methods used for Reverse Fraction Problems (N=46)

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[표 5] 문제 맥락별 정답률 (N=46) [Table 5] A percentage of correct answers according to the problem contexts

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[표 6] 이산량, 자연수 문제 맥락에서 학생들이 사용한 해결 방법 [Table 6] Students’ solution methods for the problems in which the partial quantity with a natural number is discrete

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[표 7] 연속량, 자연수 문제 맥락에서 학생들이 사용한 해결 방법 [Table 7] Students’ solution methods for the problems in which the partial quantity with a natural number is continuous

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[표 8] 연속량, 분수 문제 맥락에서 학생들이 사용한 해결 방법 [Table 8] Students’ solution methods for the problems in which the partial quantity with a fraction is continuous

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