[논문]초등학교 6학년의 분수와 소수의 크기에 대한 수직선 표상의 정확성 및 사용 전략 분석

허진영; 임수현

doi:10.7468/mathedu.2024.63.3.393

[국내논문] 초등학교 6학년의 분수와 소수의 크기에 대한 수직선 표상의 정확성 및 사용 전략 분석
Analyses of the precision and strategies for representing the magnitude of fractions and decimals on the number line among 6th graders 원문보기

Journal of the Korean Society of Mathematical Education. Series A. The Mathematical Education, v.63 no.3, 2024년, pp.393 - 409

허진영 (한양대학교) , 임수현 (한양대학교)

초록
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분수와 소수가 나타내는 크기에 대한 이해를 돕기 위해, 수의 크기를 공간상에 직관적으로 표시하는 수직선 모델이 널리 활용된다. 본 연구에서는 수직선 추정 과제를 활용하여 초등학교 6학년 학생들의 분수와 소수에 대한 이해도 및 문제 해결 전략을 살펴보고, 다양한 전략 사용의 유연성이 분수와 소수의 표상의 정확성, 연산 능력, 수학 학업성취도의 개인차와 연관성이 있는지를 분석하였다. 분석 결과, 학생들은 자연수에 비해 분수와 소수의 수직선 표상 정확성이 상대적으로 낮았으며, 특별히 분수의 경우 분모가 짝수인 분수보다 홀수인 분수에서, 소수의 경우 소수 세 자리 수 보다 소수 두 자리 수에서 수직선 표상의 정확성이 더 낮게 나타났다. 전략 사용의 측면에서 학생들은 분수를 수직선에 표상할 때 기준점 참고 전략, 분할 전략, 어림 전략 순으로 많이 사용하였으며, 소수를 표상할 때에는 기준점 참고 전략, 반올림 전략, 자연수 변환 전략 순으로 많이 사용하였다. 마지막으로 분수를 표상할 때 사용하는 전략이 다양할수록 분수의 표상 정확성과 수학 학업성취 점수가 높게 나타났다. 이러한 결과를 토대로 분수의 다양한 개념과 자연수 대비 소수의 자릿값 및 0의 개념에 대한 세심한 지도의 필요성을 제언하였다. 또한 분수와 소수에 대한 이해를 돕기 위한 방안으로, 수직선 모델 활용과 함께 표상 전략을 병행해서 지도하는 방법에 대해 논의하였다.

Abstract ▼ AI-Helper

The number line model, which intuitively marks numerical magnitudes in space, is widely utilized to help in understanding the magnitudes that fractions and decimals represent. The study analyzed 6th graders' understanding of fractions and decimals, their problem solving strategies, and whether individual differences in the flexibility of various strategy uses are associated with the accuracy of numerical representation, calculation fluency, and overall mathematical achievement. As a result of the study, students showed relatively lower accuracy in representing fractions and decimals on a number line compared to natural numbers, especially for fractions with odd denominators compared to even denominators, and for two-digit decimals compared to three-digit decimals. Regarding strategy use, students primarily used benchmark, segmentation, and approximation strategies for fractions, and benchmark, rounding, and transformation strategies for decimals sequentially. Lastly, as students used various representation strategies for fractions, their accuracy in representing fractions and their overall mathematical achievement scores showed significantly better outcomes. Taken together, we suggest the need for careful instruction on different interpretations of fractions, the place value of decimals, and the meaning of zero in decimal places. Moreover, we discuss instructional methods that integrate the number line model and its diverse representation strategies to enhance students' understanding of fractions and decimals.

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표/그림 (8)

$Figure 1. Screenshots of (a) natural, (b) fraction, and (c) decimal number line estimation tasks.$ 그림 Figure 1. Screenshots of (a) natural, (b) fraction, and (c) decimal number line estimation tasks.
$Table 1. The precision of representing fractions and decimals on the number line (measured by PAE) based on its feature$ 표 Table 1. The precision of representing fractions and decimals on the number line (measured by PAE) based on its feature
$Figure 2. The precision of representing natural numbers, fractions, decimals on the number line by items (PAE). Note. A:1/13=0.077=77 B:1/12=0.083=83 C:1/8=0.125=125 D:3/20=0.15=150 E:1/6=0.167=167 F:2/9=0.222=222 G:3/11=0.273=273 H:2/7=0.286=286 I:4/11=0.364=364 J:3/8=0.375=375 K:5/12=0.417=417 L:4/9=0.444=444 M:5/9=0.556=556 N:5/8=0.625=625 O:7/11=0.636=636 P:13/20=0.65=650 Q:5/7=0.714=714 R:8/11=0.727=727 S:5/6=0.833=833 T:17/20=0.85=850 U:7/8=0.875=875 V:8/9=0.889=889 W:10/11=0.909=909 X:13/14=0.929=929.$ 그림 Figure 2. The precision of representing natural numbers, fractions, decimals on the number line by items (PAE). Note. A:1/13=0.077=77 B:1/12=0.083=83 C:1/8=0.125=125 D:3/20=0.15=150 E:1/6=0.167=167 F:2/9=0.222=222 G:3/11=0.273=273 H:2/7=0.286=286 I:4/11=0.364=364 J:3/8=0.375=375 K:5/12=0.417=417 L:4/9=0.444=444 M:5/9=0.556=556 N:5/8=0.625=625 O:7/11=0.636=636 P:13/20=0.65=650 Q:5/7=0.714=714 R:8/11=0.727=727 S:5/6=0.833=833 T:17/20=0.85=850 U:7/8=0.875=875 V:8/9=0.889=889 W:10/11=0.909=909 X:13/14=0.929=929.
$Figure 3. The percentage of strategy selection in fraction and decimal NLE tasks.$ 그림 Figure 3. The percentage of strategy selection in fraction and decimal NLE tasks.
$Table 2. Frequency of fraction’s representation strategy selection based on numeric features (%)$ 표 Table 2. Frequency of fraction’s representation strategy selection based on numeric features (%)
표 Table 3. Frequency of decimal’s representation strategy selection based on numeric features (%)
$Table 4. The fraction PAEs, calculation test scores, and mathematical achievement test scores based on the number of strategies chosen in the fraction NLE task$ 표 Table 4. The fraction PAEs, calculation test scores, and mathematical achievement test scores based on the number of strategies chosen in the fraction NLE task
표 Table 5. Pearson correlation matrix among five variables

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