مطالعات برنامه درسی

مطالعات برنامه درسی

نقش مدل‌سازی ریاضی در ارتقای یادگیری مفاهیم ریاضی دبیرستان

نوع مقاله : مقاله پژوهشی

نویسندگان
پروین ترابی 1
شراره تقی دستجردی 2
امیر حسین اشتری 3
1 استادیار گروه ریاضی، دانشکده علوم پایه، دانشگاه صنعتی جندی شاپور دزفول، ایران
2 کارشناس ارشد ریاضی، خانه ریاضیات اصفهان، ایران.
3 کارشناس ارشد آمار، خانه ریاضیات اصفهان، ایران.
10.22034/jcs.2026.492270.2380
چکیده
هدف این پژوهش بررسی نقش مدل‌سازی ریاضی همراه با بهره‌گیری از فناوری آموزشی (نرم‌افزار صفحه‌گسترده‌ی اکسل) در یادگیری مفاهیم الگوها و دنباله‌های بازگشتی، آهنگ تغییر و کار با اعداد بزرگ است. برای این منظور دوره‌ای آموزشی مبتنی بر مدل‌سازی با محتوای مرتبط با مفاهیم مهم ریاضیات مدرسه‌ای طراحی و اجرا شد تا دانش‌آموزان بتوانند این مفاهیم را در موقعیت‌های واقعی تجربه کنند. در این مطالعه، 33 دانش‌آموز متوسطه در کارگاهی شرکت کردند که در آن با استفاده از اکسل به مدل‌سازی رشد جمعیت خرگوش‌ها پرداختند. داده‌ها از طریق پیش‌آزمون و پس‌آزمون محقق‌ساخته گردآوری و با آزمون ناپارامتری ویلکاکسون تحلیل شد. نتایج نشان داد میانگین نمرات دانش‌آموزان در هر سه متغیرِ درک دنباله‌های بازگشتی، مفهوم آهنگ تغییر و کار با اعداد بزرگ به‌طور معناداری افزایش یافته است. همچنین دانش‌آموزان پایه‌های بالاتر رشد نسبی بیشتری داشتند. یافته‌ها بیانگر آن است که مدل‌سازی ریاضی همراه با فناوری آموزشی می‌تواند زمینه‌ای مناسب برای درک عمیق‌تر مفاهیم پایه‌ی ریاضی فراهم آورد.
کلیدواژه‌ها
مدل‌سازی ریاضی
صفحه‌گسترده (اکسل)
دنباله‌های بازگشتی
آهنگ تغییر
کار با اعداد بزرگ
موضوعات

عنوان مقاله English

The Role of Mathematical Modeling in Improving High School Students’ Comprehension of Mathematical Concepts

نویسندگان English

Parvin Torabi 1
Sharareh Taghi Dastjerdi 2
Amir Hossein Ashtari 3
1 Assistant Professor, Department of Mathematics, Faculty of Basic Sciences, Jundi-Shapur University of Technology, Dezful, Iran
2 Master's degree in mathematics, Isfahan Mathematics House, Iran
3 Master's degree in Statistics, Isfahan Mathematics House, Iran
چکیده English

Introduction
Mathematical modeling provides an authentic context in which students can explore mathematical concepts through real-world phenomena. When combined with digital technologies, modeling activities can promote conceptual understanding by allowing students to investigate patterns, visualize mathematical relationships, and perform calculations that would otherwise be difficult. Among widely available educational technologies, Microsoft Excel offers a simple yet powerful environment for constructing mathematical models, generating recursive sequences, and analyzing numerical data.
This study investigates the effectiveness of an Excel-based mathematical modeling activity in improving secondary school students' understanding of three fundamental mathematical concepts: recursive patterns and sequences, rate of change, and reasoning with large numbers. The instructional activity was designed around a population-growth modeling problem that required students to formulate recursive relationships, identify emerging patterns, and interpret exponential growth using spreadsheet calculations and visual representations.
Methodology
The study employed a one-group pretest–posttest design involving 33 secondary school students from grades 9 to 11 with different educational backgrounds. The instructional intervention was implemented during two mathematics enrichment programs.
Before instruction, students completed a researcher-designed pretest assessing their understanding of recursive sequences, rate of change, and large numbers. After participating in the modeling activity using Microsoft Excel, they completed a parallel posttest measuring the same learning objectives.
Students' written responses and Excel files were analyzed using a binary scoring procedure. Minor computational or spreadsheet-entry mistakes that did not reflect conceptual misunderstanding were treated separately from genuine misconceptions. Since the paired score differences did not satisfy the normality assumption, the Wilcoxon signed-rank test was used to compare pretest and posttest performances.

Results
The results demonstrate a statistically significant improvement in students' mathematical understanding after the instructional intervention. The Wilcoxon signed-rank test revealed a significant difference between pretest and posttest scores (p < 0.05), indicating that the spreadsheet-based modeling activity positively influenced students' conceptual learning.
Students showed notable improvement in recognizing recursive patterns, interpreting rates of change, and reasoning about very large numbers generated by exponential processes. The proportion of students achieving completely correct solutions increased substantially from approximately 18% in the pretest to nearly 70% in the posttest.
Additional analyses indicated positive learning gains across different school types, grade levels, and previous modeling experience, although students with prior exposure to mathematical modeling achieved higher overall performance.

Discussion
The findings suggest that spreadsheet-supported mathematical modeling can effectively strengthen students' conceptual understanding of fundamental mathematical ideas. Rather than emphasizing routine calculations, Excel enabled students to focus on identifying mathematical structures, exploring recursive relationships, visualizing growth processes, and interpreting numerical results.
The modeling context encouraged students to connect abstract mathematical concepts with meaningful real-world situations while reducing the computational burden associated with large numerical calculations. Furthermore, the visual and interactive features of spreadsheets appeared to enhance students' engagement and confidence in solving complex mathematical problems.
Overall, this study provides empirical evidence that integrating mathematical modeling with accessible digital tools such as Microsoft Excel offers a practical and effective instructional approach for secondary mathematics education. Future research may investigate the long-term retention of conceptual understanding and compare spreadsheet-based modeling with other technology-enhanced learning environments.

کلیدواژه‌ها English

Mathematical modeling
spreadsheet (Excel)
recursive sequences
rate of change
large numbers
Ahmadpour, F., Fadaei, M. R., & Rafiepour, A. (2011). Modeling: A way to bring the real world into the classroom. In Proceedings of the First National Conference on Fundamental Transformation in Iran’s Curriculum System (pp. 516–521). Mashhad, Iran. [in Persian]
Ahmadpour, F., Taghidastjerdi, S., Rafiepour, A., Sadeghi, Z., Taheri, M., & Gharaati, E. (2023). Reality-based mathematics problems (Vol. 2). Tehran: Fatemi Publishing. [in Persian]
Ary, D., Jacobs, L. C., & Razavieh, A. (2001). Introduction to research in education (V. Sarkisian, M. Nikoo, & E. Saeedian, Trans.). Tehran: Soroush. [in Persian]
Baker, J., & Sugden, S. J. (2004). Spreadsheets in education—The first 25 years. Spreadsheets in Education, 1(1) 24-40.
Battista, M. T., Clements, D. H., Arnoff, J., Battista, K., & Van Auken Borrow, C. (1998). Students' spatial structuring of 2D arrays of squares. Journal for Research in Mathematics Education, 29(5), 503–532.
Berger, E., & Starbird, M. (2017). Five elements of effective thinking (M. A. Jafari, Trans.; 2nd ed.). Tehran: Qoqnoos Publishing.
Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events. Journal for Research in Mathematics Education, 33(5), 352–378.
 
Confrey, J., & Maloney, A. (2007). A theory of mathematical modelling in technological settings. In Modelling and applications in mathematics education: The 14th ICMI Study (pp. 57–68). Boston, MA: Springer US.
Confrey, J., & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. Educational Studies in Mathematics, 26, 135–164.
Creswell, J. W. (2014). Research Design: Qualitative, Quantitative, and Mixed Methods Approaches. Sag
Dehghani, M., & Purasfahani, V. (2025). Extracurricular activities as a communication link for quality education in multicultural classrooms: A phenomenological study in elementary school. Journal of Curriculum Studies, 20(77), 37–76. [in Persian]
Dick, T. P., & Hollebrands, K. F. (2011). Focus in high school mathematics: Technology to support reasoning and sense making (pp. xi–xvii). Reston, VA: National Council of Teachers of Mathematics.
Duval, R. (2006). A cognitive analysis of problems of comprehension in learning mathematics. Educational Studies in Mathematics, 61, 103–131.
Freudenthal, H. (1982). Research problems in mathematics education from Freudenthal’s perspective (Z. Gouya, Trans.). Roshd Journal of Mathematics Education, 65, 5–11. [in Persian]
Greer, B., & Mukhopadhyay, S. (2012). The hegemony of mathematics. In Opening the cage (pp. 227–248). Brill.
Healy, L., & Hoyles, C. (2001). Software tools for geometrical problem solving. Educational Studies in Mathematics, 44, 65–83.
Hoyles, C., & Noss, R. (2003). What can digital technologies take from and bring to research in mathematics education? Second International Handbook of Mathematics Education
Khosroshahi, L., & Asghari, A. (2016). Suitable contexts for algebra in elementary education: An analysis of Iranian textbooks based on a proposed framework. Journal of Educational Innovations, 15(3), 147–170. [in Persian]
Kaput, J. (1994). Democratizing access to calculus. Mathematics Teacher, 87, 32–36.
Mason, J. (1996). Expressing generality and roots of algebra. Educational Studies in Mathematics, 30, 65–86.
Mitchell, M. (2021). Complexity: A guided tour (R. Amirhimi, Trans.). Tehran: Nashr-e No. [in Persian]
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
National Curriculum Development Secretariat. (2010). Curriculum of the Islamic Republic of Iran: Comprehensive plan for fundamental transformation of curricula (Fourth draft). Tehran: Organization for Research and Educational Planning. [in Persian]National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.
Parhizgar, Z., Alemolhodaei, H., & Jabbari Noqabi, M. (2017). The capacity of modeling problems to change students’ attitudes toward mathematics. Theory and Practice in Curriculum Journal, 5(9), 167–192. [in Persian]
Parhizgar, Z., Jabbari Noqabi, M., & Alemolhodaei, H. (2020). The effect of teaching mathematical modeling problems on students’ joyful experiences. Journal of Educational Innovations, 19(73), 127–146. [in Persian]
Pierce, R., & Stacey, K. (2006). Enhancing the image of mathematics by association with simple pleasures from real world contexts. ZDM, 38(3), 214–225.
Pollak, H. (2007). Mathematical modelling—A conversation with Henry Pollak. In Modelling and applications in mathematics education: The 14th ICMI Study (pp. 109–120). Boston, MA: Springer US.
Rafiepour, A. (2014). Modeling and applications: A research report. Theory and Practice in Curriculum Journal, 3, 93–116. [in Persian]
Rafiepour, A., & Molayi, R. (2020). Content analysis of Iranian secondary mathematics textbooks based on a modeling approach. Research in Mathematics Education, 1(1), 29–44. [in Persian]
Rafiepour, A., & Rahmani, M. (2021). The role of problem posing in Iranian mathematics textbooks. Theory and Practice in Curriculum Journal, 17, 91–118. [in Persian]
Rivera, F. D. (2013). Teaching and learning patterns in school mathematics. New York, NY: Springer.
Sandefur, J., & Manaster, A. B. (2022). Encouraging research on recursive thinking through the lens of a model of the spread of contagious diseases. ZDM – Mathematics Education, 54(4), 895–907.
Sarama, J., & Clements, D. H. (2019). Researching and using progressions (trajectories) in mathematics education. In Global Education in the 21st Century (Vol. 3). Springer.
Secretariat of the Supreme Council of Education. (2012). Curriculum of the Islamic Republic of Iran, resolution of Esfand 2012. Tehran: Supreme Council of Education in collaboration with the Organization for Educational Research and Planning. [in Persian]
Shayan, M., & Yaftiyan, N. (2022). Evaluating the performance of ninth grade students in the math literacy test with emphasis on math textbooks. Journal of Curriculum Studies, 17(66), 41–74. [in Persian]
Supreme Council for Education Secretariat. (2012). Curriculum of the Islamic Republic of Iran: Resolution of March 2012. Tehran: Supreme Council for Education in collaboration with the Organization for Research and Educational Planning. [in Persian]
Sweller, J. (1998). Cognitive load during problem solving. Cognitive Science, 12, 257–285.
Torabi, P., Rahimi Piranfar, M., Taghidastjerdi, S., & Ashtari, A. (2024). The impact of STEM education on motivation, interest, and self-confidence in solving mathematical modeling problems using Excel. In Proceedings of the 19th National Conference on Mathematics Education in Iran. Isfahan, Iran. [in Persian]
Yousefi Hamidi, S., Salimi, L., & Fallah, V. (2025). Designing a curriculum model for economic education based on entrepreneurship for lower secondary school. Journal of Curriculum Studies, 20(77), 1–36.  [in Persian]
Yousefi, M., Zanganeh, H., Pourjamshidi, M., & Shojae, S. (2025). The role of extended reality (virtual, augmented, and mixed) in the curriculum of students with dyscalculia, dyslexia, and dysgraphia: A systematic review. Journal of Curriculum Studies, 19 (75), 53–84. [in Persian]
Ahmadpour, F., Fadaei, M. R., & Rafiepour, A. (2011). Modeling: A way to bring the real world into the classroom. In Proceedings of the First National Conference on Fundamental Transformation in Iran’s Curriculum System (pp. 516–521). Mashhad, Iran. [in Persian]
Ahmadpour, F., Taghidastjerdi, S., Rafiepour, A., Sadeghi, Z., Taheri, M., & Gharaati, E. (2023). Reality-based mathematics problems (Vol. 2). Tehran: Fatemi Publishing. [in Persian]
Ary, D., Jacobs, L. C., & Razavieh, A. (2001). Introduction to research in education (V. Sarkisian, M. Nikoo, & E. Saeedian, Trans.). Tehran: Soroush. [in Persian]
Baker, J., & Sugden, S. J. (2004). Spreadsheets in education—The first 25 years. Spreadsheets in Education, 1(1) 24-40.
Battista, M. T., Clements, D. H., Arnoff, J., Battista, K., & Van Auken Borrow, C. (1998). Students' spatial structuring of 2D arrays of squares. Journal for Research in Mathematics Education, 29(5), 503–532.
Berger, E., & Starbird, M. (2017). Five elements of effective thinking (M. A. Jafari, Trans.; 2nd ed.). Tehran: Qoqnoos Publishing.
Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events. Journal for Research in Mathematics Education, 33(5), 352–378.
 
Confrey, J., & Maloney, A. (2007). A theory of mathematical modelling in technological settings. In Modelling and applications in mathematics education: The 14th ICMI Study (pp. 57–68). Boston, MA: Springer US.
Confrey, J., & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. Educational Studies in Mathematics, 26, 135–164.
Creswell, J. W. (2014). Research Design: Qualitative, Quantitative, and Mixed Methods Approaches. Sag
Dehghani, M., & Purasfahani, V. (2025). Extracurricular activities as a communication link for quality education in multicultural classrooms: A phenomenological study in elementary school. Journal of Curriculum Studies, 20(77), 37–76. [in Persian]
Dick, T. P., & Hollebrands, K. F. (2011). Focus in high school mathematics: Technology to support reasoning and sense making (pp. xi–xvii). Reston, VA: National Council of Teachers of Mathematics.
Duval, R. (2006). A cognitive analysis of problems of comprehension in learning mathematics. Educational Studies in Mathematics, 61, 103–131.
Freudenthal, H. (1982). Research problems in mathematics education from Freudenthal’s perspective (Z. Gouya, Trans.). Roshd Journal of Mathematics Education, 65, 5–11. [in Persian]
Greer, B., & Mukhopadhyay, S. (2012). The hegemony of mathematics. In Opening the cage (pp. 227–248). Brill.
Healy, L., & Hoyles, C. (2001). Software tools for geometrical problem solving. Educational Studies in Mathematics, 44, 65–83.
Hoyles, C., & Noss, R. (2003). What can digital technologies take from and bring to research in mathematics education? Second International Handbook of Mathematics Education
Khosroshahi, L., & Asghari, A. (2016). Suitable contexts for algebra in elementary education: An analysis of Iranian textbooks based on a proposed framework. Journal of Educational Innovations, 15(3), 147–170. [in Persian]
Kaput, J. (1994). Democratizing access to calculus. Mathematics Teacher, 87, 32–36.
Mason, J. (1996). Expressing generality and roots of algebra. Educational Studies in Mathematics, 30, 65–86.
Mitchell, M. (2021). Complexity: A guided tour (R. Amirhimi, Trans.). Tehran: Nashr-e No. [in Persian]
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
National Curriculum Development Secretariat. (2010). Curriculum of the Islamic Republic of Iran: Comprehensive plan for fundamental transformation of curricula (Fourth draft). Tehran: Organization for Research and Educational Planning. [in Persian]National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.
Parhizgar, Z., Alemolhodaei, H., & Jabbari Noqabi, M. (2017). The capacity of modeling problems to change students’ attitudes toward mathematics. Theory and Practice in Curriculum Journal, 5(9), 167–192. [in Persian]
Parhizgar, Z., Jabbari Noqabi, M., & Alemolhodaei, H. (2020). The effect of teaching mathematical modeling problems on students’ joyful experiences. Journal of Educational Innovations, 19(73), 127–146. [in Persian]
Pierce, R., & Stacey, K. (2006). Enhancing the image of mathematics by association with simple pleasures from real world contexts. ZDM, 38(3), 214–225.
Pollak, H. (2007). Mathematical modelling—A conversation with Henry Pollak. In Modelling and applications in mathematics education: The 14th ICMI Study (pp. 109–120). Boston, MA: Springer US.
Rafiepour, A. (2014). Modeling and applications: A research report. Theory and Practice in Curriculum Journal, 3, 93–116. [in Persian]
Rafiepour, A., & Molayi, R. (2020). Content analysis of Iranian secondary mathematics textbooks based on a modeling approach. Research in Mathematics Education, 1(1), 29–44. [in Persian]
Rafiepour, A., & Rahmani, M. (2021). The role of problem posing in Iranian mathematics textbooks. Theory and Practice in Curriculum Journal, 17, 91–118. [in Persian]
Rivera, F. D. (2013). Teaching and learning patterns in school mathematics. New York, NY: Springer.
Sandefur, J., & Manaster, A. B. (2022). Encouraging research on recursive thinking through the lens of a model of the spread of contagious diseases. ZDM – Mathematics Education, 54(4), 895–907.
Sarama, J., & Clements, D. H. (2019). Researching and using progressions (trajectories) in mathematics education. In Global Education in the 21st Century (Vol. 3). Springer.
Secretariat of the Supreme Council of Education. (2012). Curriculum of the Islamic Republic of Iran, resolution of Esfand 2012. Tehran: Supreme Council of Education in collaboration with the Organization for Educational Research and Planning. [in Persian]
Shayan, M., & Yaftiyan, N. (2022). Evaluating the performance of ninth grade students in the math literacy test with emphasis on math textbooks. Journal of Curriculum Studies, 17(66), 41–74. [in Persian]
Supreme Council for Education Secretariat. (2012). Curriculum of the Islamic Republic of Iran: Resolution of March 2012. Tehran: Supreme Council for Education in collaboration with the Organization for Research and Educational Planning. [in Persian]
Sweller, J. (1998). Cognitive load during problem solving. Cognitive Science, 12, 257–285.
Torabi, P., Rahimi Piranfar, M., Taghidastjerdi, S., & Ashtari, A. (2024). The impact of STEM education on motivation, interest, and self-confidence in solving mathematical modeling problems using Excel. In Proceedings of the 19th National Conference on Mathematics Education in Iran. Isfahan, Iran. [in Persian]
Yousefi Hamidi, S., Salimi, L., & Fallah, V. (2025). Designing a curriculum model for economic education based on entrepreneurship for lower secondary school. Journal of Curriculum Studies, 20(77), 1–36.  [in Persian]
Yousefi, M., Zanganeh, H., Pourjamshidi, M., & Shojae, S. (2025). The role of extended reality (virtual, augmented, and mixed) in the curriculum of students with dyscalculia, dyslexia, and dysgraphia: A systematic review. Journal of Curriculum Studies, 19 (75), 53–84. [in Persian]

  • تاریخ دریافت 15 آذر 1404
  • تاریخ بازنگری 08 بهمن 1404
  • تاریخ پذیرش 29 فروردین 1405