Communications of Mathematical Education (한국수학교육학회지시리즈E:수학교육논문집)

Korean Society of Mathematical Education (한국수학교육학회)
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Formation of the function concept through change and accumulation

변화와 누적을 통한 함수 개념 형성

  • Received : 2025.12.11
  • Accepted : 2026.02.20
  • Published : 2026.03.31
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Abstract

This study investigate the quantitative processes involved in constructing relationships between two covarying quantities by examining functional thinking through the dual constructs of change and accumulation. Grounded in a covariational perspective on functions, the study aims to articulate the flow of functional reasoning that integrates these two elements and to model a corresponding learning trajectory. A design experiment was conducted with three ninth-grade students, who engaged in a task involving the filling of a rectangular bathtub at a constant rate. As students constructed a continuous relationship between time and water height, they reasoned about change using ratios and rates of change, while accumulation served as the basis for quantifying the relationship between the two quantities. The findings show that recognizing covariation is essential for quantifying continuous relationships through accumulation. Through analysis of students' reasoning, the study identifies characteristic patterns of thinking related to change and accumulation and elaborates a learning trajectory for functional thinking that incorporates these components.

본 연구에서는 함수를 공변적으로 접근할 때, 두 양 사이의 관계를 구성하는 양적인 과정을 변화와 누적이라는 두 핵심 요소를 통해 살펴본다. 이를 통해 변화와 누적이 포함된 함수적 사고의 흐름을 구체화하고, 변화와 누적을 포함한 함수적 사고의 학습경로를 모델링한다. 설계실험을 적용하여 중학교 3학년 3명의 학생을 대상으로 직육면체 모양의 욕조에 일정하게 채워지는 물의 높이를 시간에 따라 구성하는 실험을 하였다. 연구에서 학생들은 시간과 물의 높이 사이의 관계를 연속적으로 구성하는 과정에서 변화를 비와 변화율을 이용하여 추론하였다. 그리고 두 양 사이의 관계는 누적을 통해 수량화되었다. 연속적 관계를 누적으로 수량화하기 위해서는 누적에서 공변성 인식이 중요함을 확인할 수 있었다. 이를 통해 두 양의 관계를 연속적으로 구성하는 활동을 통해 함수적 사고에서 나타나는 학생들의 변화와 누적 사고 양상을 분석하고, 이를 포함하는 함수적 사고의 학습경로를 구체화하였다.

Keywords

Acknowledgement

이 논문은 한국교원대학교 2024학년도 연구년교수 학술지원비 지원을 받아 수행한 연구의 결과임.

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