بررسی تأثیر آموزش تلفیقی مدل‌سازی ریاضی و نرم‌افزار GeoGebra بر عملکرد دانش‌آموزان در یادگیری توابع مثلثاتی: یک مطالعه آمیخته

نوع مقاله : مقاله پژوهشی

نویسندگان

1 دانشجوی دکترای، آموزش ریاضی، دانشکده علوم و فناوری های همگرا، واحد علوم و تحقیقات، دانشگاه آزاد اسلامی، تهران، ایران

2 استاد، ریاضی محض، دانشکده علوم ریاضی، دانشگاه تربیت مدرس، تهران، ایران

3 دانشیار، ریاضی کاربردی، دانشکده علوم و فناوری های همگرا، واحد علوم و تحقیقات، دانشگاه آزاد اسلامی، تهران، ایران

4 دانشیار، آمار، دانشکده علوم و فناوری های همگرا، واحد علوم وتحقیقات، دانشگاه آزاد اسلامی ، تهران، ایران

چکیده

پژوهش حاضر با هدف بررسی تأثیر آموزش مبتنی بر فناوری دیجیتال (GeoGebra) و فعالیت‌های مدل‌سازی ریاضی بر عملکرد دانش‌آموزان در یادگیری توابع مثلثاتی انجام شد. این مطالعه کاربردی و کمی- کیفی با غلبه رویکرد کمی و طرح نیمه‌آزمایشی پیش‌آزمون-پس‌آزمون با دو گروه اصلی آزمایش و کنترل بود. جامعه آماری شامل دانش‌آموزان پایه یازدهم بود و نمونه نهایی ۲۷۰ نفر (۱۳۲ دانش‌آموز در 4 زیر گروه آزمایش و ۱۳۸ دانش‌آموز در 4 زیر گروه کنترل) بود. ابزار گردآوری داده‌ها شامل آزمون‌های مسئله‌محور مبتنی بر مدل‌سازی ریاضی و تحلیل عملکرد دانش‌آموزان بر اساس سطوح مهارت فنی در GeoGebra و سطوح مفهومی مدل‌سازی بود. گروه آزمایش از طریق فعالیت‌های مدل‌سازی ریاضی مبتنی بر GeoGebra آموزش دید، در حالی که گروه کنترل به روش سنتی و بدون فناوری آموزش داده شد. داده‌های کمی با پیش‌آزمون و پس‌آزمون جمع‌آوری و با تحلیل کوواریانس تک‌متغیره بررسی شدند. یافته ها نشان دادند آموزش مبتنی بر فناوری و مدل‌سازی ریاضی عملکرد دانش‌آموزان را به‌طور معنادار بهبود داد (001/0p<) با اندازه اثر بزرگ برای فناوری دیجیتال (715/0 η² =) و مدل‌سازی ریاضی (712/0η² =). مقایسه عملکرد دو مداخله تفاوت معناداری نشان نداد (339/5 = p)، که حاکی از تأثیر مثبت هر دو رویکرد است. یافته‌های کیفی نیز نشان داد دانش‌آموزان در فرآیند یادگیری فعال و تعامل گروهی، درک مفهومی عمیق‌تر و مشارکت بیشتری داشتند. در مجموع، نتایج پژوهش نشان می‌دهد ادغام فناوری دیجیتال و مدل‌سازی ریاضی می‌تواند الگویی مؤثر برای آموزش تعاملی و ارتقای عملکرد دانش‌آموزان در مباحث پیچیده ریاضی ارائه دهد.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

The Effect of Integrated Mathematical Modeling and GeoGebra on Students’ Learning of Trigonometric Functions: A Mixed-Methods Study

نویسندگان [English]

  • Samira Mehraein 1
  • Ali Iranmanesh 2
  • Mohsen Rostamy-Malkhalifeh 3
  • Mohammad Hassan Behzadi 4
1 PhD student, Department of Mathematics, Institute of Convergin Sciences and Technologies, SR.C., Islamic Azad University ,Tehran, Iran
2 Professor, Department of Mathematics, Tarbiat Modares University, Tehran, Iran
3 Associate Professor, Department of Mathematics, Institute of Convergin Sciences and Technologies, SR. C., Islamic Azad University ,Tehran, Iran
4 Associate Professor, Department of Statistics, Institute of Convergin Sciences and Technologies, SR. C., Islamic Azad University ,Tehran, Iran
چکیده [English]

The present study investigated the effect of an integrative instructional intervention combining mathematical modeling activities with GeoGebra on students’ performance in learning trigonometric functions. A quasi-experimental pretest–posttest design with two main groups (experimental and control) was employed. The final sample included 270 eleventh-grade students (132 in four experimental subgroups and 138 in four control subgroup). Data were collected using problem-based tests grounded in mathematical modeling and analyzed based on students’ technical skills in GeoGebra and conceptual modeling levels. The experimental group received GeoGebra-based mathematical modeling instruction, whereas the control group was taught using traditional methods without technology. Quantitative analyses using one-way ANCOVA indicated significant improvements in students’ performance (p < 0.001) with large effect sizes for digital technology (η² = 0.715) and mathematical modeling (η² = 0.712). Comparison of the two interventions showed no significant difference (p = 0.339), indicating the positive impact of both approaches. Qualitative findings revealed that students’ active learning, group interaction, and deeper conceptual understanding were enhanced. Overall, integrating digital technology and mathematical modeling provides an effective model for interactive instruction and improving students’ performance in complex mathematical topics.

کلیدواژه‌ها [English]

  • Integrative Instruction
  • Trigonometric Modeling
  • GeoGebra
  • Student Performance
  • Conceptual Understanding
Alqahtani, M. M., & Powell, A. B. (2016). Instrumental appropriation of a collaborative, dynamic-geometry environment and geometrical understanding. International Journal of Education in Mathematics, Science and Technology, 4(2), 72–83. https://doi.org/10.18404/ijemst.38054
Allah Karami, A (2025). The Effect of  Wordwall  Artificial Intelligence on  Agentic Engagement and Academic Performance of ElementaryStudents in Mathematics. Technology and Scholarship in Education. 5 (Special Issue): 75-91. [In Persian]
    https://doi.org/: 10.30473/t-edu.2025.75854.1332
Bilgili, S., & Çiltaş, A. (2025). Improving mathematical modeling competencies of mathematics teachers in a technology-supported learning environment. Frontiers in Education, 10, Article 1509652. https://doi.org/10.3389/feduc.2025.1509652
Borba, M. C., Chiari, A. S., & Almeida, H. R. F. L. (2025). A systematic review of technology use in middle and high school mathematics education. Frontiers in Education, 10, Article 1644284. https://doi.org/10.3389/feduc.2025.1644284
Büscher, C., & Schnell, S. (2017). Students’ emergent modeling of statistical measures: A case study. Statistics Education Research Journal, 16(2), 144–162. https://doi.org/10.52041/serj.v16i2.188
Durandt, R., Blum, W., & Lindl, A. (2021). Fostering mathematical modelling competency of South African engineering students: Which influence does the teaching design have? Educational Studies in Mathematics, 109, 361–381. https://doi.org/10.1007/s10649-021-10068-7
Darmanova, Z., Abylkassymova, A., & Nurmukhamedova, Z. (2025). A systematic review of technology use in middle and high school mathematics education: Insights from contextual, methodological, and evaluation characteristics. Frontiers in Education, 10, Article 1644284. https://doi.org/10.3389/feduc.2025.1644284
Drijvers, P., Godino, J. D., Font, V., & Trouche, L. (2013). One episode, two lenses: A reflective analysis of student learning with computer algebra from instrumental and onto-semiotic perspectives. Educational Studies in Mathematics, 82, 23–49. https://doi.org/10.1007/s10649-012-9416-8
Eckert, A., & Nilsson, P. (2022). The emergence of the “FlexTech” orchestration of inferential reasoning on pattern generalization. Digital Experiences in Mathematics Education, 8(1), 1–26. https://doi.org/10.1007/s40751-021-00098-4
Elia, I., & Spiro, R. J. (2006). Learning mathematics through multiple representations: The case of trigonometric functions. Educational Studies in Mathematics, 61(1–2), 55–74.
Engelbrecht, J., & Borba, M. C. (2023). Recent developments in using digital technology in mathematics education. ZDM–Mathematics Education, 56, 281–292. https://doi.org/10.1007/s11858-023-01530-2
Fererde, A. T., Mihrka, A. A., Ayele, M. A., & Arara, A. A. (2024). Enhancing students’ conceptual understanding and problem-solving skills in learning trigonometry through contextual-based mathematical modeling instruction. International Journal of Secondary Education, 12(4), 108–119. https://doi.org/10.11648/j.ijsedu.20241204.15
Juandi, J., Sugiatno, & Noviani, E. (2025). Enhancing students’ conceptual understanding of trigonometric functions through GeoGebra based learning. Al Jabar: Jurnal Pendidikan Matematika, 16(1), 55–70. https://doi.org/10.24042/ajpm.v16i1.25653
Koyunkaya, M. Y., & Dede, Y. (2024). Using different digital tools in designing and solving mathematical modelling problems. Education and Information Technologies, 29, 1–24. https://doi.org/10.1007/s10639-024-12577-3
Krawitz, J., Chang, Y.-P., Yang, K.-L., & Schukajlow, S. (2021). The role of reading comprehension in mathematical modelling: Improving the construction of a real-world model and interest in Germany and Taiwan. ZDM – Mathematics Education. Advance online publication.
Li, M., & Li, B. (2024). Unravelling the dynamics of technology integration in mathematics education. Education and Information Technologies, 29, 23687–23715.
Mohammadi, A., & Nasiri, M. (2025). Identifying and prioritizing the essential and fundamental requirements for smart school implementation: A case study of Khuzestan province. Journal of Education, 5(1), 9–23. [In Persian]
Moradi, R & Bayrvandi, V (2025). Effectiveness of Technology-Based Active Learning Strategy on Academic Satisfaction and Cognitive Flexibility of Students in Experimental Science Course, Technology and Scholarship in Education,4(4):41-58.
    https://doi.org/10.30473/tedu.2025.72617.1208
Mosese, N., & Ogbonnaya, U. I. (2021). GeoGebra and students' learning achievement in trigonometric functions. Cypriot Journal of Educational Sciences, 16(2), 827–846. https://doi.org/10.18844/cjes.v16i2.5685
Patel, A., & Pfannkuch, M. (2018). Developing a statistical modeling framework. ZDM–Mathematics Education, 50(7), 1197–1212.
Rojano, T., & Sutherland, R. (2020). Technology and curricula in mathematics education. In S. Lerman (Ed.), Encyclopedia of Mathematics Education (2nd ed., pp. 849–853). Springer.
Schukajlow, S., Achmetli, K., & Rakoczy, K. (2019). Does constructing multiple solutions affect self-efficacy? Educational Studies in Mathematics, 100(1), 43–60. https://doi.org/10.1007/s10649-018-9847-y
Schönbrodt, S., Wohak, K., & Frank, M. (2022). Digital tools to enable collaborative mathematical modeling online. Modelling in Science Education and Learning, 15(1), 151–174. https://doi.org/10.4995/msel.2022.16269
Firmansyah, R., Suryadi, D., & Turmudi, T. (2024). Enhancing students’ conceptual understanding and problem-solving skills in learning trigonometry through contextual-based mathematical modeling. International Journal of Secondary Education.
Freiman, V. (2020). Technology design in mathematics education. In S. Lerman (Ed.), Encyclopedia of Mathematics Education (2nd ed., pp. 853–861). Springer Nature.
Ghosh, J. B. (2015). Learning mathematics in secondary school: The case of mathematical modelling enabled by technology. In Selected regular lectures from the 12th International Congress on Mathematical Education (pp. 203–222). Springer. https://doi.org/10.1007/978-3-319-17187-6_12
Goos, M., Soury-Lavergne, S., Assude, T., Brown, J., Kong, C. M., Glover, D., & Sinclair, M. (2010). Teachers and teaching: Theoretical perspectives and issues concerning classroom implementation. In C. Hoyles & J. Lagrange (Eds.), Mathematics education and technology—Rethinking the terrain (pp. 311–328). Springer. https://doi.org/10.1007/978-1-4419-0146-0_14
Greefrath, G., Siller, H.-S., Klock, H., & Wess, R. (2022). Pre-service secondary teachers’ pedagogical content knowledge for the teaching of mathematical modelling. Educational Studies in Mathematics, 109(2), 383–407. https://doi.org/10.1007/s10649-021-10038-z
Gurmu, F., Tuge, C., & Hunde, A. B. (2024). Effects of GeoGebra assisted instructional methods on students’ conceptual understanding of geometry. Cogent Education, 11(1), 2379745. https://doi.org/10.1080/2331186X.2024.2379745
Haugstad, A., & Isabwe, G. M. N. (2017). Using the theory of instrumental genesis to study students’ work with a digital tool for applying integrals in a kinematic simulation. In Proceedings of CERME 10 (pp. 2088–2096). Dublin, Ireland.
Hochmuth, R., Peters, J., Rønning, F., & Winsløw, C. (2024). Modelling mathematics for educational research and practice: A comparison of two theoretical approaches. Educational Studies in Mathematics, 118, 153–168.
Van Dijke-Droogers, M., Drijvers, P., & Bakker, A. (2021). Statistical modeling processes through the lens of instrumental genesis. Educational Studies in Mathematics, 107(2), 235–260. https://doi.org/10.1007/s10649-020-10023-y
Siregar, T. (2025). Integrating GeoGebra in Mathematics Education: Enhancing Pedagogical Practices among Teachers and Lecturers.Preprints. https://doi.org/10.20944/preprints202510.1359.v1
Usman, A. N., Arifudin, R., Ilham Aditya, R., Maulana, B. S., & Lavicza, Z. (2024). Using different digital tools in designing and solving mathematical modelling problems. Education and Information Technologies, 29, 19035–19065.
  • تاریخ دریافت: 08 آذر 1404
  • تاریخ بازنگری: 20 بهمن 1404
  • تاریخ پذیرش: 29 فروردین 1405
  • تاریخ اولین انتشار: 25 خرداد 1405
  • تاریخ انتشار: 01 مهر 1405