Authors : Dr. Tamashwar Budhoo; Dannyram Seawpersaud

Volume/Issue : Volume 11 - 2026, Issue 9 - September

Google Scholar : https://tinyurl.com/4au5wuyx

DOI : https://doi.org/10.38124/ijisrt/26sep1322

Note : A published paper may take 4-5 working days from the publication date to appear in PlumX Metrics, Semantic Scholar, and ResearchGate.

Abstract : George Pólya’s four-phase model of problem solving, which comprises understanding the problem, devising a plan, carrying out the plan, and looking back, remains one of the most widely cited heuristics in mathematics education. This conceptual paper examines the extent to which the model, as it is commonly enacted in secondary school curricula and classrooms, accounts for problem solving as contemporary research describes it. The paper distinguishes Pólya’s original heuristic programme from the reductive checklist into which it has frequently been transformed, and it directs its critique principally at the latter. Adopting a theory synthesis design, the paper examines six dimensions: the recursive rather than linear character of problem solving, the continuous role of metacognitive regulation, the diversity of learners’ conceptual, linguistic, and affective resources, the mediating influence of digital technology, the socially mediated nature of classroom problem solving, and the place of creativity and insight. The paper also engages counterarguments from cognitive load theory, which support explicit and structured guidance for novices. The synthesis concludes that Pólya’s model is not a historical artefact; rather, it offers a sound foundation that requires reconceptualisation. The paper therefore proposes a recursive, regulated, and mediated framework in which Pólya’s phases form an iterative core, metacognitive regulation operates across all phases, and social, digital, linguistic, and affective conditions shape the process. The paper closes with implications for classroom practice, teacher education, curriculum and assessment design, and future empirical research.

Keywords : Problem Solving; Pólya; Heuristics; Metacognition; Socio-Constructivism; Digital Technology; Mathematical Creativity; Secondary Mathematics Education.

References :

1. P. Liljedahl and J. Cai, “Empirical research on problem solving and problem posing: A look at the state of the art,” ZDM – Mathematics Education, vol. 53, no. 4, pp. 723–735, 2021, doi: 10.1007/s11858-021-01291-w.
2. M. Santos-Trigo, “Problem solving in mathematics education: Tracing its foundations and current research-practice trends,” ZDM – Mathematics Education, vol. 56, pp. 211–222, 2024, doi: 10.1007/s11858-024-01578-8.
3. G. Pólya, How to Solve It: A New Aspect of Mathematical Method. Princeton, NJ, USA: Princeton University Press, 1945.
4. B. Rott, “Pólya’s influence on (my) research,” Teaching Mathematics and Computer Science, vol. 20, no. 2, pp. 161–171, 2022, doi: 10.5485/TMCS.2022.0530.
5. A. H. Schoenfeld, “Thoughts on Pólya’s legacy,” Teaching Mathematics and Computer Science, vol. 20, no. 2, pp. 157–160, 2022, doi: 10.5485/TMCS.2022.0541.
6. G. Pólya, Mathematics and Plausible Reasoning, vols. 1–2. Princeton, NJ, USA: Princeton University Press, 1954.
7. A. H. Schoenfeld, “Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics,” in Handbook of Research on Mathematics Teaching and Learning, D. A. Grouws, Ed. New York, NY, USA: Macmillan, 1992, pp. 334–370.
8. A. H. Schoenfeld, Mathematical Problem Solving. Orlando, FL, USA: Academic Press, 1985.
9. M. P. Carlson and I. Bloom, “The cyclic nature of problem solving: An emergent multidimensional problem-solving framework,” Educational Studies in Mathematics, vol. 58, no. 1, pp. 45–75, 2005, doi: 10.1007/s10649-005-0808-x.
10. B. Rott, B. Specht, and C. Knipping, “A descriptive phase model of problem-solving processes,” ZDM – Mathematics Education, vol. 53, no. 4, pp. 737–752, 2021, doi: 10.1007/s11858-021-01244-3.
11. J. Garofalo and F. K. Lester, “Metacognition, cognitive monitoring, and mathematical performance,” Journal for Research in Mathematics Education, vol. 16, no. 3, pp. 163–176, 1985.
12. J. Engelbrecht and M. C. Borba, “Recent developments in using digital technology in mathematics education,” ZDM – Mathematics Education, vol. 56, no. 2, pp. 281–292, 2024, doi: 10.1007/s11858-023-01530-2.
13. M. Goos, P. Galbraith, and P. Renshaw, “Socially mediated metacognition: Creating collaborative zones of proximal development in small group problem solving,” Educational Studies in Mathematics, vol. 49, no. 2, pp. 193–223, 2002, doi: 10.1023/A:1016209010120.
14. E. Jaakkola, “Designing conceptual articles: Four approaches,” AMS Review, vol. 10, no. 1, pp. 18–26, 2020, doi: 10.1007/s13162-020-00161-0.
15. L. Verschaffel, S. Schukajlow, J. Star, and W. Van Dooren, “Word problems in mathematics education: A survey,” ZDM – Mathematics Education, vol. 52, no. 1, pp. 1–16, 2020, doi: 10.1007/s11858-020-01130-4.
16. L. S. Vygotsky, Mind in Society: The Development of Higher Psychological Processes. Cambridge, MA, USA: Harvard University Press, 1978.
17. R. Borromeo Ferri, “Theoretical and empirical differentiations of phases in the modelling process,” ZDM, vol. 38, no. 2, pp. 86–95, 2006, doi: 10.1007/BF02655883.
18. J. H. Flavell, “Metacognition and cognitive monitoring: A new area of cognitive–developmental inquiry,” American Psychologist, vol. 34, no. 10, pp. 906–911, 1979, doi: 10.1037/0003-066X.34.10.906.
19. M. V. J. Veenman, B. H. A. M. Van Hout-Wolters, and P. Afflerbach, “Metacognition and learning: Conceptual and methodological considerations,” Metacognition and Learning, vol. 1, no. 1, pp. 3–14, 2006, doi: 10.1007/s11409-006-6893-0.
20. B. J. Zimmerman, “Becoming a self-regulated learner: An overview,” Theory Into Practice, vol. 41, no. 2, pp. 64–70, 2002, doi: 10.1207/s15430421tip4102_2.
21. Z. R. Mevarech and B. Kramarski, “IMPROVE: A multidimensional method for teaching mathematics in heterogeneous classrooms,” American Educational Research Journal, vol. 34, no. 2, pp. 365–394, 1997, doi: 10.3102/00028312034002365.
22. B. Kramarski, Z. R. Mevarech, and M. Arami, “The effects of metacognitive instruction on solving mathematical authentic tasks,” Educational Studies in Mathematics, vol. 49, no. 2, pp. 225–250, 2002, doi: 10.1023/A:1016282811724.
23. T. Tambychik and T. S. M. Meerah, “Students’ difficulties in mathematics problem-solving: What do they say?,” Procedia – Social and Behavioral Sciences, vol. 8, pp. 142–151, 2010, doi: 10.1016/j.sbspro.2010.12.020.
24. P. Phonapichat, S. Wongwanich, and S. Sujiva, “An analysis of elementary school students’ difficulties in mathematical problem solving,” Procedia – Social and Behavioral Sciences, vol. 116, pp. 3169–3174, 2014, doi: 10.1016/j.sbspro.2014.01.728.
25. J. Abedi and C. Lord, “The language factor in mathematics tests,” Applied Measurement in Education, vol. 14, no. 3, pp. 219–234, 2001, doi: 10.1207/S15324818AME1403_2.
26. G. Ramirez, H. Chang, E. A. Maloney, S. C. Levine, and S. L. Beilock, “On the relationship between math anxiety and math achievement in early elementary school: The role of problem solving strategies,” Journal of Experimental Child Psychology, vol. 141, pp. 83–100, 2016, doi: 10.1016/j.jecp.2015.07.014.
27. P. Drijvers, “Digital technology in mathematics education: Why it works (or doesn’t),” PNA, vol. 8, no. 1, pp. 1–20, 2013, doi: 10.30827/pna.v8i1.6120.
28. A. Kuzle, “Nature of metacognition in a dynamic geometry environment,” LUMAT: International Journal on Math, Science and Technology Education, vol. 3, no. 5, pp. 627–646, 2015, doi: 10.31129/lumat.v3i5.1010.
29. F. Barrera-Mora and A. Reyes-Rodríguez, “Cognitive processes developed by students when solving mathematical problems within technological environments,” The Mathematics Enthusiast, vol. 10, no. 1–2, pp. 109–136, 2013.
30. S. J. Hegedus and Y. Otálora, “Mathematical strategies and emergence of socially mediated metacognition within a multi-touch dynamic geometry environment,” Educational Studies in Mathematics, vol. 112, no. 2, pp. 289–307, 2023, doi: 10.1007/s10649-022-10170-4.
31. J. M. Wing, “Computational thinking,” Communications of the ACM, vol. 49, no. 3, pp. 33–35, 2006, doi: 10.1145/1118178.1118215.
32. D. Weintrop, E. Beheshti, M. Horn, K. Orton, K. Jona, L. Trouille, and U. Wilensky, “Defining computational thinking for mathematics and science classrooms,” Journal of Science Education and Technology, vol. 25, no. 1, pp. 127–147, 2016, doi: 10.1007/s10956-015-9581-5.
33. D. Wood, J. S. Bruner, and G. Ross, “The role of tutoring in problem solving,” Journal of Child Psychology and Psychiatry, vol. 17, no. 2, pp. 89–100, 1976, doi: 10.1111/j.1469-7610.1976.tb00381.x.
34. P. Cobb and E. Yackel, “Constructivist, emergent, and sociocultural perspectives in the context of developmental research,” Educational Psychologist, vol. 31, no. 3–4, pp. 175–190, 1996, doi: 10.1080/00461520.1996.9653265.
35. E. Yackel, P. Cobb, and T. Wood, “Small-group interactions as a source of learning opportunities in second-grade mathematics,” Journal for Research in Mathematics Education, vol. 22, no. 5, pp. 390–408, 1991.
36. N. M. Webb, “Task-related verbal interaction and mathematics learning in small groups,” Journal for Research in Mathematics Education, vol. 22, no. 5, pp. 366–389, 1991, doi: 10.2307/749186.
37. M. Walshaw and G. Anthony, “The teacher’s role in classroom discourse: A review of recent research into mathematics classrooms,” Review of Educational Research, vol. 78, no. 3, pp. 516–551, 2008, doi: 10.3102/0034654308320292.
38. E. A. Silver, “Fostering creativity through instruction rich in mathematical problem solving and problem posing,” ZDM, vol. 29, no. 3, pp. 75–80, 1997, doi: 10.1007/s11858-997-0003-x.
39. E. L. Mann, “Creativity: The essence of mathematics,” Journal for the Education of the Gifted, vol. 30, no. 2, pp. 236–260, 2006, doi: 10.4219/jeg-2006-264.
40. R. Leikin and D. Pitta-Pantazi, “Creativity and mathematics education: The state of the art,” ZDM – The International Journal on Mathematics Education, vol. 45, no. 2, pp. 159–166, 2013, doi: 10.1007/s11858-012-0459-1.
41. P. A. Kirschner, J. Sweller, and R. E. Clark, “Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching,” Educational Psychologist, vol. 41, no. 2, pp. 75–86, 2006, doi: 10.1207/s15326985ep4102_1.
42. F. K. Lester, “Musings about mathematical problem-solving research: 1970–1994,” Journal for Research in Mathematics Education, vol. 25, no. 6, pp. 660–675, 1994, doi: 10.2307/749576.
43. F. K. Lester, “Thoughts about research on mathematical problem-solving instruction,” The Mathematics Enthusiast, vol. 10, no. 1–2, pp. 245–278, 2013, doi: 10.54870/1551-3440.1267.
44. M. Kapur, “Productive failure in learning math,” Cognitive Science, vol. 38, no. 5, pp. 1008–1022, 2014, doi: 10.1111/cogs.12107.
45. A. H. Schoenfeld, “Pólya, problem solving, and education,” Mathematics Magazine, vol. 60, no. 5, pp. 283–291, 1987, doi: 10.1080/0025570X.1987.11977325.