مطالعات برنامه درسی

مطالعات برنامه درسی

واکاوی رویکردهای معلمان ریاضی به ابزارهای روانشناختی درتدریس مفهوم معادله: یک مطالعه پدیدارنگاری

نوع مقاله : مقاله پژوهشی

نویسندگان
1 گروه آموزش ریاضی، دانشگاه فرهنگیان، صندوق پستی 889- 14665 تهران، ایران
2 بخش آموزش ریاضی، دانشکده ریاضی و کامپیوتر، دانشگاه شهید باهنر، کرمان، ایران و پژوهشکده ریاضی ماهانی، پژوهشگاه افضلی پور، دانشگاه
چکیده
هدف این پژوهش، تبیین نقش ابزارهای روان‌شناختی در آموزش مفهوم تساوی و معادله و پر کردن شکاف میان نظریه‌های میانجی‌گری شناختی و عمل تربیتی در کلاس درس است. مطالعه با رویکرد کیفی و پدیدارنگاری و با مشارکت چهارده معلم ریاضی پایه هفتم در شهر کرمان که با روش نمونه‌گیری هدفمند انتخاب شدند، انجام شد. داده‌ها از طریق مصاحبه‌های نیمه‌ساختاریافته گردآوری و با روش دالگرن و فالسبرگ تحلیل شدند؛ روایی با بازبینی مشارکت‌کنندگان و پایایی با مقایسه‌ی متخصص پدیدارنگاری تأیید شد. یافته‌ها نشان دادند که معلمان از چهار دسته ابزار میانجی‌گر شامل زبان، حرکات بدنی، نمادها و مصنوعات آموزشی بهره می‌گیرند. زبان در سه نقش فعال‌سازی ذهن، هدایت پله‌ای و تثبیت استعاری معنا ظاهر شد؛ حرکات بدنی پل میان تجربه‌ی حسی و نمادین بودند؛ نمادها روابط جبری را تثبیت کردند و مصنوعات کمتر استفاده شدند. نتیجه‌گیری نشان داد آموزش معادله فرایندی چندوجهی و نشانه‌ای است. نوآوری پژوهش در ترکیب رویکرد پدیدارنگاری با چارچوب نشانه‌شناختی و تمرکز بر تجربه‌ی معلمان نهفته است.
کلیدواژه‌ها
موضوعات

عنوان مقاله English

Analyzing Mathematics Teachers' Approaches to Psychological Tools in Teaching the Concept of Equation: A Phenomenographic Study

نویسندگان English

fatemeh Mirzaei 1
Abolfazl Rafiepour 2
1 1 Department of Mathematics Education, Farhangian University, P.O. Box 14665- 889, Tehran, Iran
2 Department of Mathematics Education, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran & Mahani Math Center, Afzalipour Research Institute, Shahid Bahonar University of Kerman, Iran.
چکیده English

The aim of this study is to explain the role of psychological tools in teaching the concept of equality and equation and to bridge the gap between theories of cognitive mediation and educational practice in the classroom. The study was conducted using a qualitative phenomenographic approach with the participation of fourteen seventh‑grade mathematics teachers in Kerman, selected through purposive sampling. Data were collected through semi‑structured interviews and analyzed using the Dahlgren and Fallsberg method; validity was ensured through participant review and reliability was confirmed by comparison with a phenomenography specialist. Findings showed that teachers used four categories of mediating tools: language, bodily movements, symbols, and educational artifacts. Language appeared in three roles: activating thinking, guiding step by step, and stabilizing meaning through metaphors; bodily movements acted as a bridge between sensory and symbolic experience; symbols consolidated algebraic relations; and artifacts were used less frequently. The conclusion indicated that teaching equations is a multifaceted and semiotic process. The novelty of this study lies in combining phenomenography with a semiotic framework and focusing on teachers’ experiences.

کلیدواژه‌ها English

Concept of equality
equation
mathematics teacher
psychological tools
phenomenography
Abtahi, Y. (2016). Things kids think with: The role of the physical properties of mathematical tools in children’s learning in the context of addition of fractions [Doctoral dissertation, University of Ottawa]. ResearchUOttawa. https://doi.org/10.20381/RUOR-223
Amini-Far, A., & Karimi-Kia, Kh. (2014). Enhancing students’ learning in the first order equation. Journal of Educational Sciences, Shahid Chamran University of Ahvaz, 21(2), 23–38. [In Persian]
Bajwa, N., & Perry, M. (2021). Features of a pan balance that may support students’ developing understanding of mathematical equivalence. Mathematical Thinking and Learning, 21(1), 1–27. https://doi.org/10.1080/10986065.2020.1700587 
Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective. In L. D. English (Ed.), Handbook of international research in mathematics education (2nd ed., pp. 746–783). Routledge.
Bosque, J. S. (2025). Common Challenges' Students Face When Learning Linear Equation. International Journal of Multidisciplinary Research and Analysis, 8(4), 2100–2115. https://doi.org/ 10.47191/ijmra/v8-i04-65
Carpenter, T., Franke, M., & Levi, L. (2003). Thinking mathematically. Heinemann. Filloy, E., & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19–25.
Dahlgren, L. O., and Fallsberg, M. (1991). Phenomenography as a qualitative approach in social pharmacy research. Journal of social and administrative pharmacy, 8(4): 150-156.
Faramarzpur, N., & Fadaei, M. R. (2021). Mathematics Teachers' Opinions on the Causes of Educational Mistakes of Student in School Classrooms with an Emphasis on Algebra. Journal of Theory & Practice in Curriculum, 9(17), 395–428.
Flick, U. (2006). An introduction to qualitative research. Translated by H. Jalili, 1404). Ney Publishing. [In Persian]
Goya, Z., Fadaei, M. R., & Agah, Z. (2013). Mathematics Teachers’ Approaches to “Listening” in Mathematics Classes: A phenomenographic approach. Curriculum Theory and Practice, 1(1), 27–48
Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12(3), 317–326. https://doi.org/10.1007/BF00311062
Maffia, A., & Mariotti, M. A. (2018). From using artefacts to mathematical meanings: The teacher’s role in the semiotic mediation process. Didattica della matematica. Dalla ricerca alle pratiche d’aula, 4, 50–63. https://doi.org/10.33683/ddm.18.4.3.1
Magruder, R., & Mohr-Schroeder, M. J. (2013). Investigating the use of virtual pan balances in elementary classrooms. Journal of Computers in Mathematics and Science Teaching, 32(4), 455–478.
Marton, F. (1981). Phenomenography—Describing conceptions of the world around us. Instructional Science, 10(2), 177–200.
Maschietto, M. (2015). Teachers, students and resources in mathematics laboratory. In S. J. Cho (Ed.), Selected Regular Lectures from the 12th International Congress on Mathematical Education (ICME‑12) (pp. 527–546). Springer International Publishing. https://doi.org/10.1007/978-3-319-17187-6_30
Matthews, P., & Rittle-Johnson, B. (2009). In pursuit of knowledge: Comparing self-explanations, concepts, and procedures as pedagogical tools. Journal of Experimental Child Psychology, 104(1), 1–21.  https://doi.org/10.1016/j.jecp.2008.08.004
McNeil, N., Grandau, L., Knuth, E., Alibali, M., Stephens, A., Hattikudur, S., & Krill, D. (2006). Middle-school students’ understanding of the equal sign: The books they read can’t help. Cognition and Instruction, 24(3), 367–385. https://doi.org/10.1207/s1532690xci2403_3
Meira, L. (1998). Making sense of instructional devices: the emergence of transparency in mathematical activity. Journal for Research in Mathematics Education, 29(2), 121-142.  https://doi.org/10.2307/749895
Niroo, M. (2025). A Model of the Hidden Curriculum in Mathematics Education Based on a Socio-cultural Approach. Journal of Curriculum Studies, 20(77), 77–100. https://doi.org/  10.22034/jcs.2025.471290.2317
Noss, R., & Hoyles, C. (1996). Windows on mathematical meanings: Learning cultures and computers (Vol. 17). New York, USA: Springer
Ntšohi, M. M. E. (2005). An investigation into the problems encountered by learners and teachers of Grade 9 algebra on understanding linear equations: A critical analysis (Unpublished master's dissertation). University of South Africa. http://uir.unisa.ac.za/bitstream/handle/10500/2509/00front.pdf?sequence=1
Núñez, R. (2008). Gestures, diagrams, and the mathematical mind. In L. English (Ed.), Handbook of International Research in Mathematics Education (2nd ed., pp. 357–380). New York: Routledge.
Radford, L. (2003). Gestures, Speech, and the Sprouting of Signs: A Semiotic-Cultural Approach to Students’ Types of Generalization. Mathematical Thinking and Learning, 5(1), 37–70. https://doi.org/10.1207/S15327833MTL0501_02
Radford, L. (2002). The seen, the spoken and the written. A semiotic approach to the problem of objectification of mathematical knowledge. For the Learning of Mathematics, 22(2), 14–23.
Radford, L. (2009). Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings. Educational Studies in Mathematics, 70(2), 111–126. https://doi.org/10.1007/s10649-008-9127-33
Radford, L. (2022). Introducing equations in early algebra. ZDM – Mathematics Education, 54(6), 1151–1167.  https://doi.org/10.1007/s11858-022-01422-x
Reeve, R. A., & Reynolds, F. J. (2001). The role of gesture in co-constructing mathematical understanding. In D. Clarke (Ed.), Perspectives on practice and meaning in mathematics and science classrooms (pp. 231–253). Dordrecht: Kluwer Academic Publishers.
Rittle-Johnson, B., Matthews, P., & Taylor, R. (2011). Assessing knowledge of mathematical equivalence: A construct-modeling approach. Journal of Educational Psychology, 103(1), 85–104. http://doi.org/10.1037/a0021334
Robutti, O., Sabena, C., Krause, C., Soldano, C., & Arzarello, F. (2022). Gestures in Mathematics Thinking and Learning. In M. Danesi (Ed.), Handbook of Cognitive Mathematics. Cham: Springer.
Sfard, A. & Linchevski, L. (2010). The story of algebra: Advantages and pitfalls of reification (Z. Kamyab & A. H. Asghari, Trans.). Roshd-e Amouzeshe Riazi, 27(3), 4–12; 28(1), 13–19; 28(2), 17–25.
Stacey, K. & Asghari, A. H. (2009). Moving from arithmetic thinking to algebraic thinking. Roshd-e Amouzeshe Riazi [Mathematics Education Journal], 26(3), 3–11. [In Persian].
Stacey, K. & Vincent, J. (2009). Modes of reasoning in explanations in Australian eighth-grade mathematics textbooks. Educational Studies in Mathematics, 72(2), 271–288.  http://doi.org/10.1007/s10649-009-9193-1
Schliemann, A. D., Carraher, D.W., Goodrow, A., Caddle, M. C., & Porter, M. (2013). Equations in Elementary School, Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education, Lindmeier, A. M. & Heinze, A., (Eds.); Kiel, Germany: PME. 4, 161-168.
Stephens, A., Knuth, E., Blanton, M., Isler, I., Gardiner, A., & Marum, T. (2013). Equation structure and the meaning of the equal sign: The impact of task selection in eliciting elementary students’ understandings. Journal of Mathematical Behavior, 32(2), 173–182. http://doi.org/10.1016/j.jmathb.2013.02.001 
Stephens, A., Sung, Y., Strachota, S., Torres, R., Morton, K., Gardiner, A., Blanton, M., Knuth, E., & Stroud, R. (2022). The role of balance scales in supporting productive thinking about equations among diverse learners. Mathematical Thinking and Learning, 24(1), 1–18. https://doi.org/10.1080/10986065.2020.1793055
Suh, J. M., & Moyer-Packenham, P. S. (2007). Developing students’ representational fluency using virtual manipulatives. Mathematics Teaching in the Middle School, 13(3), 208–216.
Tossavainen, T., Attorps, I., & Väisänen, P. (2011). On Mathematics Students Understanding of the Equation Concept. Journal of Far East Mathematical Education, 6 (2), 127-147. https://doi.org/10.17654/0973563111013
Van de Pol, J., Volman, M., & Beishuizen, J. (2010). Scaffolding in teacher–student interaction: A decade of research. Educational Psychology Review, 22(3), 271–296. https://doi.org/10.1007/s10648-010-9127-6
Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press.
Vygotsky, L. S. (1981). The genesis of higher mental functions. In J. V. Wertsch (Ed.), The concept of activity in Soviet psychology (pp. 134–143). Armonk, NY: Sharpe.
Vygotsky, L. S. (1997). The collected works of L. S. Vygotsky, Vol. 4: The history of the development of higher mental functions (R. W. Rieber, Ed.). New York: Plenum Press.
Wertsch, J. V. & Addison Stone, C. (1995). The concept of internalization in Vygotsky’s account of the genesis of higher mental functions. In J. V. Wertsch (Ed), Culture, communication and cognition: Vygotskian perpsectives. Cambridge University Press.
Wood, D., Bruner, J. S., & Ross, G. (1976). The role of tutoring in problem solving. Journal of Child Psychology and Psychiatry, 17(2), 89–100.
دوره 21، شماره 81 - شماره پیاپی 81
با تمرکز بر جنبه های فرهنگی و اجتماعی برنامه درسی
تابستان 1405
صفحه 155-178

  • تاریخ دریافت 06 دی 1404
  • تاریخ بازنگری 02 اسفند 1404
  • تاریخ پذیرش 05 اردیبهشت 1405