[논문]단위 조정 단계가 다른 중학생의 이차함수 문제 해결 과정에서 나타나는 특징

이진아; 이수진

doi:10.7468/mathedu.2022.61.3.441

단위 조정 단계가 다른 중학생의 이차함수 문제 해결 과정에서 나타나는 특징
A case study on the quadratic function problem solving process of middle school students with different unit coordination stages 원문보기

Journal of the Korean Society of Mathematical Education. Series A. The Mathematical Education, v.61 no.3, 2022년, pp.441 - 456

이진아 (광교중학교) , 이수진 (한국교원대학교)

초록
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본 연구의 목적은 학생들의 단위 조정과 학교 수학과의 관계를 이해하기 위한 목적으로 수행되고 있는 프로젝트의 일부 결과를 보고하는 것이다. 구체적으로 단위 조정 단계와 그에 따른 수준이 다른 학생들이 y = ax² 형태인 이차함수 문제를 해결하는데 있어 비례 지식이 어떻게 사용되고, 단위 조정 수준별 가용한 지식은 무엇인지 세밀하게 분석하는 것이다. 이를 위해 자연수 맥락에서는 3수준 단위를 주어진 자원으로 사용하여 동화할 수 있는 단위 조정 3단계 학생이지만, 복잡한 분수 곱셈 과제에서는 서로 다른 단위 조정 단계를 보여준 중학교 1학년 세 학생에 초점을 두었다. 나아가 비례 문제 해결 과정과 비례 관계가 포함된 이차함수 관련 문제에 대한 임상 면담 자료를 분석하였다. 분석 결과, 단위 조정 단계에 따라 비례 문제를 해결하는 과정에서 학생들의 지식은 다르게 나타났으며, 이러한 차이는 이차함수를 이해하고 식으로 표현하는 과정에서 결과적 차이를 보였다. 이러한 분석 결과를 통해 결론에서는 단위 조정 이론, 비례 지식, 그리고 이차함수 지식과의 관련성에 대해 논의 후 시사점을 제시하였다.

Abstract ▼ AI-Helper

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.

주제어

표/그림 (11)

그림 Figure 1. Two perspectives on the missing value problem situation using proportional relations (Carney et al., 2015, p.142).
표 Table 1. Steinthorsdottir and Sriraman (2009)'s understanding of scalar and functional perspectives on the level of proportional reasoning (Carney et al., 2015, p. 143).
$Figure 2. Sangwoo's partitioning from eye measurements and Dahye's images for <TEX>$\frac{4}{3}$</TEX> of 11.$ 그림 Figure 2. Sangwoo's partitioning from eye measurements and Dahye's images for $\frac{4}{3}$ of 11.
$Figure 3. Yeonsu's picture of the <TEX>$\frac{4}{3}$</TEX> situation on <TEX>$\frac{4}{3}$</TEX>.$ 그림 Figure 3. Yeonsu's picture of the $\frac{4}{3}$ situation on $\frac{4}{3}$.
표 Table 2. Interview tasks.
그림 Figure 4. The process of solving the missing value problem using Sangwoo's proportional relationship.
그림 Figure 5. The process of solving the missing value problem using Dahye's proportional relationship.
그림 Figure 6. A table numerically showing the vertical length and the area of the rectangle relative to the horizontal length of Sangwoo, Dahye, Yeonsu (from left).
그림 Figure 7. Algebraic expression for a quadratic function of Dahye showing a quantitative operation similar to a proportional problem solving process.
그림 Figure 8. Flexible transition between scalar and functional perspectives in Yeonsu's problem-solving process.
그림 Figure 9. An explanatory hypothesis for the quadratic problem solving process from the proportional problem of three students.

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