Scaffolding ako didaktický prostriedok v riešiteľskom procese matematických úloh
Scaffolding as a Didactic Tool in the Problem-Solving Process of Mathematical Problems
Abstract
The article examines the use of scaffolding as a supportive tool in the process of solving (not only) mathematical problems. It presents selected didactic applications that illustrate how purposefully formulated questions and procedures can differentiate the level of support provided to students and foster their understanding. The empirical part is based on the analysis of a word problem from the external part of the mathematics graduation examination, demonstrating the potential of scaffolding to intervene at individual phases of the problem-solving process – from understanding the problem to reflection, verification, and searching for alternative solving approaches (looking back). The findings suggest that the thoughtful integration of scaffolding, including its lateral forms, can enhance students' problem-solving competencies, support their metacognitive processes, and foster conditions for adaptive and productive mathematical learning.
- Keywords
- scaffolding, laterálny scaffolding, riešiteľský proces, riešiteľské stratégie
- Classification
- 97D50, 97C30
References
Amiripour, P., Amir-Mofidi, S., & Shahvarani, A. (2012). Scaffolding as effective method for mathematical learning. Indian Journal of Science and Technology, 5(9), 1–4. 3328-3331. Available from: https://doi.org/10.17485/ijst/2012/v5i9.6 [viewed 2025-11-22].
Applebaum, M., Leikin, R. (2007). Looking back at the beginning: Critical thinking insolving unrealistic problems. The Montana Mathematics Enthusiast, 4(2), 258–26.
Bakker, A., Smit, J., & Wegerif, R. (2015). Scaffolding and dialogic teaching in mathematics education: introduction and review. ZDM, 47(7), 1047–1065. Available from: https://doi.org/10.1007/s11858-015-0738-8 [viewed 2025-11-22].
Buttuler, M. (2025). EEF Scaffolding in the Classroom: How to Use and Fade Support [online]. (14. novembra 2025) Available from: https://www.oltinternational.net/blogs/blog/eef-scaffolding-fade-support-primary-secondary [viewed 2025-11-22].
Firnanda, G. I., & Susiswo. (2024). Analysis of students’ dual processes in solving number pattern questions and providing scaffolding. Journal Focus Action of Research Mathematic (Factor M), 7(2), 258–275. Available from: https://doi.org/10.30762/f_m.v7i2.3499 [viewed 2025-11-22].
Isnania, I., Huda, N., & Hariyanto, H. (2021). Scaffolding on student construction errors in mathematics problem solving. Indonesian Journal of Science and Mathematics Education, 4(3), 313–323. Available from: https://doi.org/10.24042/ijsme.v4i3.10173 [viewed 2025-11-22].
Jannah, U. R., Saleh, H., & Wahidah, A. (2019). Scaffolding untuk Pembelajaran Matematika di Kelas Inklusi. JKPM (Jurnal Kajian Pendidikan Matematika), 5(1), Available from: https://doi.org/10.30998/jkpm.v5i1.5254 [viewed 2025-11-22].
Klerlein, J., Hervey, S. (2019). Mathematics as a Complex Problem-Solving Activity: Promoting Students’ Thinking Through Problem-Solving. Generation Ready White Paper. Available online: https://generationready.com/wp-content/uploads/2025/03/Mathematics-as-a-Complex-Problem-Solving-Activity.pdf [viewed 2025-11-22].
Kusmaryono, I., Gufron, A. M., & Rusdiantoro, A. (2020). Effectiveness of scaffolding strategies in learning against decrease in mathematics anxiety level. NUMERICAL Jurnal Matematika Dan Pendidikan Matematika, 13–22. Available from: https://doi.org/10.25217/numerical.v4i1.770 [viewed 2025-11-22].
Lajoie, S. P. (2005). Extending the scaffolding metaphor. Instructional Science, 33(5–6), 541–557
Martin, N. D., Tissenbaum, C. D., Gnesdilow, D., & Puntambekar, S. (2018). Fading distributed scaffolds: the importance of complementarity between teacher and material scaffolds. Instructional Science, 47(1), 69–98. Available from: https://doi.org/10.1007/s11251-018-9474-0 [viewed 2025-11-22].
McLeod, S. (2025). Vygotsky's Zone of Proximal Development. In Simply Psychology. Available from: https://www.simplypsychology.org/zone-of-proximal-development.html [viewed 2025-11-22].
Murdiyani, N. M. (2013). Scaffolding to Support Better Achievement in Mathematics. PYTHAGORAS : Jurnal Matematika dan Pendidikan Matematika, 8(1), 84-91. Available from: https://doi.org/10.21831/pg.v8i1.8496 [viewed 2025-11-22].
Nance, A. (2024). 7 Scaffolding learning strategies for the classroom. University of San Diego - Professional & Continuing Education. Available from: https://pce.sandiego.edu/scaffolding-in-education-examples/ [viewed 2025-11-22].
Nugroho, P. (2017). Scaffolding meningkatkan kemampuan berpikir kritis dalam pembelajaran matematika. Jurnal Silogisme Kajian Ilmu Matematika Dan Pembelajarannya, 2(1), 15. Available from: DOI:10.24269/JS.V2I1.500 [viewed 2025-11-22].
Nurfadila, N. R. A., Utami, N. D. M., Im, N. F. a. J., & Rahmawati, N. P. (2024). Enhancing Algebraic Problem Solving Through Scaffolding: a Case Study of Indonesian Junior High School Students. Proceeding of International Conference on Education and Sharia, 1, 700–707. Available from: https://doi.org/10.62097/ices.v124.108 [viewed 2025-11-22].
NIVAM (2018a). Maturita 2018 – Externá časť. (kód testu 7070). Available from: https://www2.nucem.sk/dl/741/RT_MAT_7070_2018.pdf [viewed 2025-11-23].
NIVAM (2018b). Správa o výsledkoch riadneho termínu externej časti maturity z matematiky. Available from: https://www2.nucem.sk/dl/3843/MAT_-Spr%C3%A1va_o_v%C3%BDsledkoch_E%C4%8C_MS_2018_final.pdf [viewed 2025-11-23].
NIVAM (2019). PISA 2018, Národná správa, Slovensko. Available from:
https://www2.nucem.sk/dl/4636/Narodna_sprava_PISA_2018.pdf [viewed 2025-11-23].
NIVAM (2023). PISA 2022. Available from:
https://www2.nucem.sk/dl/5944/tvorive%20myslenie_KRATKA_SPRAVA_final.pdf [viewed 2025-11-23].
Piaget, J., & Inhelder. B. (1997). Psychologie dítěte. Praha: Portál, 1997, 144 s.
Polya, G. (1957). How To Solve It. Second Edition. Princeton University Press, New Yersey.
Rachmawati, L. N., Sah, R. W. A., Hasanah, S. N., & Hazarika, A. (2023). Newman and scaffolding stages in analyzing student errors in solving algebraic problems. Delta-Phi Jurnal Pendidikan Matematika, 1(1), 01–05. Available from: https://doi.org/10.61650/dpjpm.v1i1.30 [viewed 2025-11-22].
Salam, M., Ndia, L., Misu, L., & Jafar, J. (2024). Applying scaffolding technique in problem based learning model on students’ mathematics problem solving ability. European Journal of Educational and Development Psychology, 12(3), 50–63. Available from: https://doi.org/10.37745/ejedp.2013/vol12n35063 [viewed 2025-11-22].
Srinivas, P., & Gireesh, D. S (2025). A Comprehensive Analysis of Lateral Scaffolding Techniques in Enhancing Student Math Problem Solving. International Journal of Mathematical Education, 15(1). ISSN 0973-6948, 7-15. Available from: https://dx.doi.org/10.37622/IJoME/15.1.2025 [viewed 2025-11-22].
Vygotsky, L. S. (1978). Mind in Society: The Development of Higher Psychological Processes. Cambridge, MA: Harvard University Press.
Wheeler, S. (2022). Learning theories: Jerome Bruner on the Scaffolding of Learning. In TeachThought. Available from: https://www.teachthought.com/learning/jerome-bruner/ [viewed 2025-11-22].
Wood, D. J., Bruner, J. S., & Ross, G. (1976). The role of tutoring in problem solving. Journal of Child Psychology and Psychiatry, 17(2), 89-100. Available from: https://doi.org/10.1111/j.1469-7610.1976.tb00381.x [viewed 2025-11-22].
Zeng, W. (2024). An exploration of difficult Problem-Solving Strategies in Inquiry-Based Learning teaching Practices based on the Creation Orientation Perspective—Taking the secondary school math curriculum as an example. Advances in Education, 14(08), 1169–1174. Available from: https://doi.org/10.12677/ae.2024.1481537 [viewed 2025-11-22].
Zumbach, J., Reimann, P. (2003). Influence of Feedback on Distributed Problem Based Learning. In: Wasson, B., Ludvigsen, S., Hoppe, U. (eds) Designing for Change in Networked Learning Environments. Computer-Supported Collaborative Learning, vol 2. Springer, Dordrecht. Available from: https://doi.org/10.1007/978-94-017-0195-2_28 [viewed 2025-11-22].
Applebaum, M., Leikin, R. (2007). Looking back at the beginning: Critical thinking insolving unrealistic problems. The Montana Mathematics Enthusiast, 4(2), 258–26.
Bakker, A., Smit, J., & Wegerif, R. (2015). Scaffolding and dialogic teaching in mathematics education: introduction and review. ZDM, 47(7), 1047–1065. Available from: https://doi.org/10.1007/s11858-015-0738-8 [viewed 2025-11-22].
Buttuler, M. (2025). EEF Scaffolding in the Classroom: How to Use and Fade Support [online]. (14. novembra 2025) Available from: https://www.oltinternational.net/blogs/blog/eef-scaffolding-fade-support-primary-secondary [viewed 2025-11-22].
Firnanda, G. I., & Susiswo. (2024). Analysis of students’ dual processes in solving number pattern questions and providing scaffolding. Journal Focus Action of Research Mathematic (Factor M), 7(2), 258–275. Available from: https://doi.org/10.30762/f_m.v7i2.3499 [viewed 2025-11-22].
Isnania, I., Huda, N., & Hariyanto, H. (2021). Scaffolding on student construction errors in mathematics problem solving. Indonesian Journal of Science and Mathematics Education, 4(3), 313–323. Available from: https://doi.org/10.24042/ijsme.v4i3.10173 [viewed 2025-11-22].
Jannah, U. R., Saleh, H., & Wahidah, A. (2019). Scaffolding untuk Pembelajaran Matematika di Kelas Inklusi. JKPM (Jurnal Kajian Pendidikan Matematika), 5(1), Available from: https://doi.org/10.30998/jkpm.v5i1.5254 [viewed 2025-11-22].
Klerlein, J., Hervey, S. (2019). Mathematics as a Complex Problem-Solving Activity: Promoting Students’ Thinking Through Problem-Solving. Generation Ready White Paper. Available online: https://generationready.com/wp-content/uploads/2025/03/Mathematics-as-a-Complex-Problem-Solving-Activity.pdf [viewed 2025-11-22].
Kusmaryono, I., Gufron, A. M., & Rusdiantoro, A. (2020). Effectiveness of scaffolding strategies in learning against decrease in mathematics anxiety level. NUMERICAL Jurnal Matematika Dan Pendidikan Matematika, 13–22. Available from: https://doi.org/10.25217/numerical.v4i1.770 [viewed 2025-11-22].
Lajoie, S. P. (2005). Extending the scaffolding metaphor. Instructional Science, 33(5–6), 541–557
Martin, N. D., Tissenbaum, C. D., Gnesdilow, D., & Puntambekar, S. (2018). Fading distributed scaffolds: the importance of complementarity between teacher and material scaffolds. Instructional Science, 47(1), 69–98. Available from: https://doi.org/10.1007/s11251-018-9474-0 [viewed 2025-11-22].
McLeod, S. (2025). Vygotsky's Zone of Proximal Development. In Simply Psychology. Available from: https://www.simplypsychology.org/zone-of-proximal-development.html [viewed 2025-11-22].
Murdiyani, N. M. (2013). Scaffolding to Support Better Achievement in Mathematics. PYTHAGORAS : Jurnal Matematika dan Pendidikan Matematika, 8(1), 84-91. Available from: https://doi.org/10.21831/pg.v8i1.8496 [viewed 2025-11-22].
Nance, A. (2024). 7 Scaffolding learning strategies for the classroom. University of San Diego - Professional & Continuing Education. Available from: https://pce.sandiego.edu/scaffolding-in-education-examples/ [viewed 2025-11-22].
Nugroho, P. (2017). Scaffolding meningkatkan kemampuan berpikir kritis dalam pembelajaran matematika. Jurnal Silogisme Kajian Ilmu Matematika Dan Pembelajarannya, 2(1), 15. Available from: DOI:10.24269/JS.V2I1.500 [viewed 2025-11-22].
Nurfadila, N. R. A., Utami, N. D. M., Im, N. F. a. J., & Rahmawati, N. P. (2024). Enhancing Algebraic Problem Solving Through Scaffolding: a Case Study of Indonesian Junior High School Students. Proceeding of International Conference on Education and Sharia, 1, 700–707. Available from: https://doi.org/10.62097/ices.v124.108 [viewed 2025-11-22].
NIVAM (2018a). Maturita 2018 – Externá časť. (kód testu 7070). Available from: https://www2.nucem.sk/dl/741/RT_MAT_7070_2018.pdf [viewed 2025-11-23].
NIVAM (2018b). Správa o výsledkoch riadneho termínu externej časti maturity z matematiky. Available from: https://www2.nucem.sk/dl/3843/MAT_-Spr%C3%A1va_o_v%C3%BDsledkoch_E%C4%8C_MS_2018_final.pdf [viewed 2025-11-23].
NIVAM (2019). PISA 2018, Národná správa, Slovensko. Available from:
https://www2.nucem.sk/dl/4636/Narodna_sprava_PISA_2018.pdf [viewed 2025-11-23].
NIVAM (2023). PISA 2022. Available from:
https://www2.nucem.sk/dl/5944/tvorive%20myslenie_KRATKA_SPRAVA_final.pdf [viewed 2025-11-23].
Piaget, J., & Inhelder. B. (1997). Psychologie dítěte. Praha: Portál, 1997, 144 s.
Polya, G. (1957). How To Solve It. Second Edition. Princeton University Press, New Yersey.
Rachmawati, L. N., Sah, R. W. A., Hasanah, S. N., & Hazarika, A. (2023). Newman and scaffolding stages in analyzing student errors in solving algebraic problems. Delta-Phi Jurnal Pendidikan Matematika, 1(1), 01–05. Available from: https://doi.org/10.61650/dpjpm.v1i1.30 [viewed 2025-11-22].
Salam, M., Ndia, L., Misu, L., & Jafar, J. (2024). Applying scaffolding technique in problem based learning model on students’ mathematics problem solving ability. European Journal of Educational and Development Psychology, 12(3), 50–63. Available from: https://doi.org/10.37745/ejedp.2013/vol12n35063 [viewed 2025-11-22].
Srinivas, P., & Gireesh, D. S (2025). A Comprehensive Analysis of Lateral Scaffolding Techniques in Enhancing Student Math Problem Solving. International Journal of Mathematical Education, 15(1). ISSN 0973-6948, 7-15. Available from: https://dx.doi.org/10.37622/IJoME/15.1.2025 [viewed 2025-11-22].
Vygotsky, L. S. (1978). Mind in Society: The Development of Higher Psychological Processes. Cambridge, MA: Harvard University Press.
Wheeler, S. (2022). Learning theories: Jerome Bruner on the Scaffolding of Learning. In TeachThought. Available from: https://www.teachthought.com/learning/jerome-bruner/ [viewed 2025-11-22].
Wood, D. J., Bruner, J. S., & Ross, G. (1976). The role of tutoring in problem solving. Journal of Child Psychology and Psychiatry, 17(2), 89-100. Available from: https://doi.org/10.1111/j.1469-7610.1976.tb00381.x [viewed 2025-11-22].
Zeng, W. (2024). An exploration of difficult Problem-Solving Strategies in Inquiry-Based Learning teaching Practices based on the Creation Orientation Perspective—Taking the secondary school math curriculum as an example. Advances in Education, 14(08), 1169–1174. Available from: https://doi.org/10.12677/ae.2024.1481537 [viewed 2025-11-22].
Zumbach, J., Reimann, P. (2003). Influence of Feedback on Distributed Problem Based Learning. In: Wasson, B., Ludvigsen, S., Hoppe, U. (eds) Designing for Change in Networked Learning Environments. Computer-Supported Collaborative Learning, vol 2. Springer, Dordrecht. Available from: https://doi.org/10.1007/978-94-017-0195-2_28 [viewed 2025-11-22].