[논문]Freudenthal의 교수학적 현상학에 기반한 일차함수 개념 수학화 과정 사례 분석

김은숙; 조완영

doi:10.7468/mathedu.2022.61.3.419

Freudenthal의 교수학적 현상학에 기반한 일차함수 개념 수학화 과정 사례 분석
Mathematising process analysis of linear function concept based on Freudenthal's didactical phenomenology 원문보기

Journal of the Korean Society of Mathematical Education. Series A. The Mathematical Education, v.61 no.3, 2022년, pp.419 - 439

김은숙 (남성중학교) , 조완영 (충북대학교)

초록
AI-Helper

본 연구의 목적은 프로이덴탈의 수학화 과정과 일차함수 개념의 교수학적 현상학의 분석을 통하여 학생들이 변화율이 일정한 현상을 표, 그래프, 식으로 표현하는 과정과 일차함수 개념의 심상을 구성하는 과정, 본질을 구성하는 과정을 서술하고 분석하는 것이다. 연구 결과, 학생들은 변화율이 일정한 현상을 표로 표현할 때 합성단위로서의 비를 사용하고, 그래프로 표현할 때 한 학생을 제외하고 직선으로 표현하였다. 식으로 표현할 때 학생별로 주어진 상황, 공변 관점, 대응 규칙을 이용하는 수준의 차이가 있었다. 학생들은 두 변화량 사이의 관계를 곱셈적으로 비교하였고, 그 비율이 하나의 상수가 된다는 것을 교사의 안내에 따라 구성하였다. 특히 시간의 변화량과 거리의 변화량이 하나의 값, 속력이 되는 상황을 통하여 변화율이 일정하다는 심상을 구성하였다. 단, 변화율을 직선의 기울기와 연결하는 데에는 어려움을 겪었으나, 변화율이 일정하다는 심상과 그래프가 직선이며 식의 모양이 y=ax+b (a는 변화율, b는 절편)라는 심상을 정리하고 조직하여 일차함수의 본질(개념)을 구성하였다.

Abstract ▼ AI-Helper

This study is based on Freudenthal's mathmatising process and the didactical phenomenology of linear function concept, I have described and examined the process in which students represent the constant rate of change into tables, graphs and equations and, in this way, how they construct mental objects and essence of the linear function concept. The students used the proportionality as composite units, when they represented the phenomenon with constant rate of change into tables. When representing in graphs, all but one student represented it into a line. There were differences among the students in the level they were using the given conditions, co-variation perspective, and corresponding rules when formulating equations. The students compared the relationship between two variables in a multiplicative way, and under the guidance of teachers they reached to the understanding that its relationship becomes a constant. Moreover, they could construct mental objects of a constant rate of change, understanding the situation where the relationship between time difference and distance difference becomes one value, namely speed. The students had difficulties in connecting the rate of change with the inclination of a line. The students constructed the essence (concept) of linear functions, after building and organizing the image that the rate of change is constant, the graph is linear, and the equation is formulated as y=ax+b (a: inclination, b: intercept).

주제어

표/그림 (9)

그림 Figure 1. Freudenthal's process of mathematisation in concepts.
그림 Figure 2. Defnition of slope (Kim et al., 2019)
그림 Figure 3. Uniform motion's expression and analysis (Kim et al., 2020, pp. 96-97).
그림 Figure 4. Linear function concept's essence.
표 Table 1. Developed linear function concept curriculum.
표 Table 2. Analysis of Students' mathematization process for linear function concept.
그림 Figure 5. Student A's graph.
그림 Figure 6. Student D's graph
그림 Figure 7. Student A's graph 2.

참고문헌 (25)

Chong, Y.O. (1997). Sudy on Fredudenthal's mathematising instruction theory [Doctoral dissertation, Seoul National Unversity Graduate School].
Chong, Y.O., Lee, K.H., Na, G.S. (2018). Realistic mathematics education, Kyowoo.
Ellis, A. (2013). Teaching ratio and proportion in the middle grades: Ratio and proportion. Research Brief. National Council of Teachers of Mathematics.
Gravemeijer, K., Stephan. M., Julie, C., Lin, F-L., & Ohtani, M. (2017). What mathematics education may prepare students for the future. International Journal of Science and Mathematics Education, 15(1), 105-123. https://doi.org/10.1007/s10763-017-9814-6

상세보기
Na, G.S., Kim, D.W., Kim, Y., Lee, S.J., Park, M.M. (2017), Future Mathematics Education's view and research. The Korea Society of Educational Studies in Mathematics.
Joanne Lobato & Amy B. Ellis(Park, J.S., Kang, H.Y., Ko, E.S., Lee, D.H. Transfer) (2016). Developing Essential Understanding of Ratios, Proportions, and Proportional Reasoning for Teaching Mathematics, Kyowoosa.
Kaput, James J. (1994). "The Representational roles of technology in connecting mathematics with authentic experience", Didactics of Mathematics as a Scientific Discipline, Kluwer Academic Publishers. 379-397.
Kim, N.H., Na, G.S., Park, K.M., Lee, K.H., Chong, Y.O., Hong, J.G. (2017). A Study on Mathematics Curriculum and Textbooks , Gyeongmunsa.
Kim, H.K., Na, G.S., Lee, M.L., Lee, A.K., Kwon, Y.G. (2019). Middle School Mathematics 2, Good book Shinsago.
Kim, H.R., Kim S.H., Kim, M.S., Lee, Y.S., Hwang, S.Y., Lee, S.H., ..., Song, S.J. (2020). Middle School Science 3, DongA Press.
Kim, Y.U. & Kim, Y.G. (1986). Mathematical history, Woosung Munwhasa..
Kim, Y.U. & Kim, Y.G. (1999). Understanding of mathematical history, Shin Sung Press.
Ma, M.Y. (2017). Middle School Students' Understanding and Development of Linear Functions [Doctor's thesis, Korea National University of Education Graduate School].
National Council of Teachers of Mathematics (2010) (Ryu, H.C., Cho, W.Y., Lee, K.H., Na, G.S., Kim, N.G., Pang, J.S. Transfer) (2007). Principles and Rules for School Mathematics, Gyeongmunsa.
Park, J.H., Shin, J.H., Lee, S.J., Ma, M.Y. (2017). Analyzing Students' Works with Quantitative and Qualitative Graphs Using Two Frameworks of Covariational Reasoning, Jounal of Educational Research in Mathematics, 27(1), 23-49.
Ryou, H.J. (2008). A Study of Conceptual Understanding and Teaching Methods for of Linear Function Slope [Master's thesis, Korea National University of Education Graduate School].
Seo, J.S. (2009). The Analysis of Understanding the Slope Concept in Hing School Students [Master's thesis, Korea National University of Education Graduate School].
Shin, S.J. & Cho, W.Y. (2020). An Analysis of the Concepts of Direct Proportion in Textbook Under 2015- Revised Curriculum Based on Freudenthal's Mathematising Instruction Theory, School Mathematics, 22(4), 923-943.
Small, Christopher G. (2007). Functional Equations and How to Solve Them, Springer.
The Ministry of Education. (2015). Mathematics Curriculum. Ministry of Education , 2015-74 [Supplement 8].
Thomas J. Cooney, Sybilla Beckmann, Gwendolyn M. Lloyd(Cho, W.Y., Kwon, N.Y., Lee, D.H. Trasfer) (2017), Developing Essential Understanding of Functions for Teaching Mathematics, Kyowoosa.
Thompson, P.W., & Carlson, M.P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for research in mathematics education (pp.421-456). National Council of Teachers of Mathematics.
Vinner, S (1991). The Role of Definitions in the Teaching and Learning of Mathematics, Advanced Mathematical Thinking, 65-81., Springer https://doi.org/10.1007/0-306-47203-15
Woo, J.H. (2017). School Mathematics's Educational Basic(2), Seoul National Unversity Press.
Yi, G.H. & Lee, J.H. (2020). Pre-service Teachers' Ways of Thinking of Qualitative Graph Construction in a Continuous Covariation Situation, Jounal of Educational Research in Mathematics, 30(3), 509-530. https://doi.org/10.29275/jerm.2020.08.30.3.509

상세보기

내보내기 구분	파일저장 인쇄 메일전송
구성항목	기본정보 상세정보 관리번호, 논문명, 저널/프로시딩명, 저자 , 발행년, 권, 호, 시작페이지, 끝페이지, 발행기관 관리번호, 논문명, 대등논문명, 저자 , 저널/프로시딩명, 발행기관, 발행년, 발행언어, 권, 호, 시작페이지, 끝페이지, ISBN, ISSN, 주제분야, 키워드, 초록(한글), 초록(영문), 저자(소속기관)
저장형식	Text(ASCII format) Excel format RefWorks Direct Export RIS format (for Reference Manager, ProCite, EndNote), Scholar's Aids, Mendeley
메일정보	받는사람 (필수) @ 보내는사람 (선택) @ 제목 내용 KISTI 검색결과 이메일 서비스
안내	총 건의 자료가 검색되었습니다. 다운받으실 자료의 인덱스를 입력하세요. (1-10,000) 검색결과의 순서대로 최대 10,000건 까지 다운로드가 가능합니다. 데이타가 많을 경우 속도가 느려질 수 있습니다.(최대 2~3분 소요) 다운로드 파일은 UTF-8 형태로 저장됩니다. 파일의 내용이 제대로 보이지 않을실 때는 웹브라우저 상단의 보기 -> 인코딩 -> 자동선택 여부를 확인하십시오. ~ Text(ASCII format) Excel format

연합인증