Communications of Mathematical Education (한국수학교육학회지시리즈E:수학교육논문집)

Korean Society of Mathematical Education (한국수학교육학회)
권호 표지
DOI QR코드

DOI QR Code

Analysis of covariational reasoning in technology-based mathematics lessons

공학 도구 활용 수업에서 학생들의 공변 추론 분석

  • Kim, Yuri (Graduate school of Chonnam National University) ;
  • Shin, Bomi (Chonnam National University)
  • Received : 2026.02.11
  • Accepted : 2026.03.10
  • Published : 2026.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

The purpose of this study is to analyze the characteristics and developmental patterns of middle school students' covariational reasoning as they explore variables through dragging in a classroom environment using AlNuSet, a dynamic algebraic engineering tool. To this end, dynamic variable-based algebra tasks were designed based on the "Change and Relationship" strand of the 2022 revised Korean mathematics curriculum and implemented in five class sessions with 26 first-year middle school students. Students' verbal interactions and records of their manipulations in AlNuSet were collected during the lessons and qualitatively analyzed using the covariational reasoning framework proposed by Thompson and Carlson. The results show that students' covariational reasoning appeared in diverse forms, ranging from qualitative recognition of change to coordination of individual values, reasoning based on units of change, and further to the recognition of invariant relationships that hold across the entire process of change. In particular, Level 4 covariational reasoning did not emerge as a fixed or isolated stage, but rather appeared in combination with other levels and functioned as an important cognitive stepping stone for transitions from qualitative reasoning to more advanced forms of covariational reasoning. In addition, some students, without direct teacher guidance, autonomously adjusted the scope of their exploration by using features such as domain settings and tracking functions, thereby extending their reasoning toward the accumulation of change units and invariant relationships. These findings suggest that dynamic algebra tasks using AlNuSet support students in conceptualizing variables not as fixed unknowns but as varying quantities, and facilitate the progressive development of covariational reasoning. Furthermore, this study provides concrete instructional design implications for early algebra instruction in middle school, highlighting how covariational reasoning can serve as a central means for exploring connections between algebra and functions.

본 연구의 목적은 동적 대수 공학도구 알누셋을 활용한 교수 실험에 참여한 중학교 1학년 학생들의 공변 추론 양상과 수업 맥락에서 나타나는 수준 변화를 분석하여 대수와 함수를 통합적으로 다루는 교수·학습 방안을 모색하는 데 있다. 이를 위해 2022 개정 교육과정 '변화와 관계' 영역의 '방정식과 그 해' 단원을 중심으로 26명을 대상으로 5차시 수업을 실시하였다. 수집된 발화 및 조작 자료를 Thompson과 Carlson(2017)의 공변 추론 수준에 비추어 분석한 결과, 학생들은 질적 변화 인식에서 출발하여 개별 값 대응과 변화 단위 기반 판단을 거쳐 두 양의 불변 관계를 전체 범위에서 인식하는 수준으로까지 변화가 드러났다. 이는 공학 도구를 활용하여 변수를 변하는 양으로 다루는 경험이 상위 수준의 공변 추론을 지원함을 시사하며, 초기 대수에서 대수와 함수의 통합적 지도에 있어 의미있는 시사를 준다.

Keywords

References

  1. 교육부. (2022). 수학과 교육과정. 교육부 고시 제2022-33호 [별책 8]. Ministry of Education. (2022). Mathematics curriculum, Ministry of Education Notice No. 2022-33 [Annex 8].
  2. 김채연, 신재홍. (2016). 연속적으로 공변하는 두 양에 대한 추론의 차이가 문제 해결에 미치는 영향. 수학교육, 55(3), 251-279. https://doi.org/10.7468/mathedu.2016.55.3.251 Kim, C. Y., & Shin, J. H. (2016). The effect of differences in reasoning about two continuously covarying quantities on problem solving. The Mathematical Education, 55(3), 251-279. https://doi.org/10.7468/mathedu.2016.55.3.251
  3. 마민영. (2021). 두 중학생의 공변 추론 수준에 따른 연립방정식 문장제의 해결에서 나타나는 유사성 비교. 수학교육 논문집, 35(3), 323-340. https://doi.org/10.7468/jksmee.2021.35.3.323 Ma, M. Y. (2021). Comparison of similarities in solving systems of linear equations word problems according to two middle school students' covariational reasoning levels. Communications of Mathematical Education, 35(3), 323-340. https://doi.org/10.7468/jksmee.2021.35.3.323
  4. 마민영. (2023). 대수 문장제를 해결하고 일반화하는 과정에서 드러난 두 중학생의 공변 추론 수준 비교. 수학교육 논문집, 37(4), 569-590. https://doi.org/10.7468/jksmee.2023.37.4.569 Ma, M. Y. (2023). Comparison of middle school students' similarities revealed in the process of word problems solving according to covariational reasoning. Communications of Mathematical Education, 37(4), 569-590. https://doi.org/10.7468/jksmee.2023.37.4.569
  5. 박종희. (2019). 중학교 1학년 학생들의 비례 추론과 공변 추론의 관계 분석 [박사학위논문, 한국교원대학교 대학원]. Park, J. H. (2019). Relationship between proportional reasoning and covariational reasoning of the 7th grade [Doctoral dissertation, Korea National University of Education].
  6. 박종희, 신재홍, 이수진, 마민영. (2017). 그래프 유형에 따른 두 공변 추론 수준 이론의 적용 및 비교. 수학교육학연구, 27(1), 23-49. 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. The Journal of Educational Research in Mathematics, 27(1), 23-49.
  7. 방정숙, 김리나, 곽은애. (2023). 초등학교 5학년 학생들의 일반화된 산술 관점과 함수적 관점에서의 변수에 대한 이해. 수학교육 논문집, 37(3), 419-442. https://doi.org/10.7468/jksmee.2023.37.3.419 Pang, J. S., Kim, L., & Kwak, E. A. (2023). Fifth graders' understanding of variables from a generalized arithmetic and a functional perspectives. Communications of Mathematical Education, 37(3), 419-442. https://doi.org/10.7468/jksmee.2023.37.3.419
  8. 양성현. (2021). GeoGebra를 활용한 연립이차방정식 교수·학습 방안 연구. East Asian Mathematical Journal, 37(2), 265-288. https://doi.org/10.7858/eamj.2021.018 Yang, S. H. (2021). A study on teaching and learning methods of simultaneous quadratic equations using GeoGebra. East Asian Mathematical Journal, 37(2), 265-288. https://doi.org/10.7858/eamj.2021.018
  9. 오서영. (2024). 영재학교 학생의 GeoGebra를 활용한 과정 중심 평가 수행과제 탐구. 한국학교수학회논문집, 27(2), 241-255. http:s//doi.org/10.30807/ksms.2024.27.2.007 Oh, S. Y. (2024). Exploring gifted school students' process-based assessment performance using GeoGebra: Focused on the quadratic curves. Journal of the Korean School Mathematics Society, 27(2), 241-255. http:s//doi.org/10.30807/ksms.2024.27.2.007
  10. 오세준. (2024). 고등학교 수학 교과서의 공학 도구 활용 현황 분석. 수학교육 논문집, 38(2), 263-286. https://doi.org/10.7468/jksmee.2024.38.2.263 Oh, S. J. (2024). An analysis of the use of technology tools in high school mathematics textbooks based. Communications of Mathematical Education, 38(2), 263-286. https://doi.org/10.7468/jksmee.2024.38.2.263
  11. 우정호, 김성준. (2007). 대수의 사고 요소 분석 및 학습-지도 방안의 탐색. 수학교육학연구, 17(4), 453-475. Woo, J. H., & Kim, S. J. (2007). Analysis of the algebraic thinking factors and search for the direction of its learning and teaching. Journal of Educational Research in Mathematics, 17(4), 453-475.
  12. 최다혜. (2013). GeoGebra를 활용한 이차함수 그래프 지도에 대한 사례 연구: 고등학교 1학년 수학 학습부진아를 대상으로 [석사학위논문, 아주대학교]. Choi, D. H. (2013). A case study on teaching quadratic function graphs using GeoGebra: Focusing on low-achieving 10th-grade students [Master's thesis, Ajou University].
  13. 황재우. (2019). 공변 추론 신장을 위한 수행과제 개발 및 적용과정에서의 사고분석 [석사학위논문, 한국교원대학교]. Hwang, J. W. (2019). Application of performance task for improving middle school students' covariational reasoning [Master's thesis, Korea National University of Education].
  14. 하수현, 이광호. (2011). 초등학교 6학년 학생들의 변수 개념 이해에 관한 사례 연구. 수학교육, 50(2), 213-231. Ha, S.-H., & Lee, K.-H. (2011). Case study on the sixth graders' understanding of concepts of variable. Mathematics Education, 50(2), 213-231.
  15. Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practises in Cabri environments. ZDM-Mathematics Education, 34(3), 66-72. https://doi.org/10.1007/BF02655708
  16. Balomenou, A., Komis, V., & Zacharos, K. (2017). Handling signs in inequalities by exploiting multiple dynamic representations—The case of AlNuSet. Digital Experiences in Mathematics Education, 3, 39-69. https://doi.org/10.1007/s40751-017-0029-9
  17. Bose, A. (2014). Revisiting transfer of learning in mathematics: Insights from an urban low-income settlement. In C. Nicol, P. Liljedahl, S. Oesterle, & D. Allan (Eds.) Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education and the 36th Conference of PME-NA (Vol. 2, pp. 177-184). PME.
  18. Carlson, M. P., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. J ournal for Research in Mathematics Education, 33(5), 352-378. https://doi.org/10.2307/4149958
  19. Chiappini, G., & Pedemonte, B. (2010). The design of new digital artefacts as a key factor to innovate the teaching and learning of algebra: The case of AlNuSet. In V. Durand-Guerrier et al. (Eds.), Proceedings of CERME 6 (WG7), (pp.1389-1398) INRP.
  20. Chiappini, G., & Pedemonte, B. (2012). Dynamic and visuospatial techniques mediated by technology for the algebraic activity. Electronic Journal of Mathematics & Technolog, 6(2), 141-158.
  21. Chiappini, G., Pedemonte, B., & Robotti, E. (2008). Using Alnuset to construct the notions of equivalence and equality in algebra. In J. Voogt & G. Knezek (Eds.), Learning to live in the knowledge society (IFIP WCC TC3 2008) (pp. 345-348). Springer. https://doi.org/10.1007/978-0-387-09729-9_50
  22. Chiappini, G., Robotti, E., & Trgalova, J. (2010). Role of an artefact of dynamic algebra in the conceptualisation of the algebraic equality. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of CERME 6 (pp. 619-628). INRP.
  23. Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Reidel Pub. Co.
  24. Leung, A. (2008). Dragging in a dynamic geometry environment through the lens of variation. International Journal of Computers for Mathematical Learning, 13(2), 135-157. https://doi.org/10.1007/s10758-008-9130-x
  25. Leung, A. (2011). An epistemic model of task design in dynamic geometry environment. ZDM Mathematics Education, 43(3), 325-336. https://doi.org/10.1007/s11858-011-0329-2
  26. Leung, A. (2016). Exploring techno-pedagogic task design in the mathematics classroom. In A. Leung (Ed.), Digital technologies in designing mathematics education tasks (pp. 3-16). https://doi.org/10.1007/978-3-319-43423-0_1
  27. Marton, F. (2014). Necessary conditions of learning. Routledge. https://doi.org/10.4324/9781315816876
  28. National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Author.
  29. Robotti, E. (2016). Designing innovative learning activities to face difficulties in algebra of dyscalculic students: Exploiting the functionalities of AlNuSet. In A. Leung & A. Baccaglini-Frank (Eds.), Digital technologies in designing mathematics education tasks (pp. 193-214). Springer. https://doi.org/10.1007/978-3-319-43423-0_10
  30. Smith, J. P., & Thompson, P. W. (2007). Quantitative reasoning and the development of algebraic reasoning. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 95-132). Erlbaum.
  31. Thompson, P. W. (1996). Imagery and the development of mathematical reasoning. In L. P. Steffe, P. Nesher, P. Cobb, G. A. Goldin, & B. Greer (Eds.), Theories of learning mathematical (pp. 267-283). Erlbaum.
  32. 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.
  33. Vamvakoussi, X., Van Dooren, W., & Verschaffel, L. (2013). Educated adults are still affected by intuitions about the effect of arithmetical operations: Evidence from a reaction-time study. Educational Studies in Mathematics, 82(2), 323-330. https://doi.org/10.1007/s10649-012-9432-8