Research Article

A model for scaffolding mathematical problem-solving: From theory to practice

Yong Khin Tay 1 , Tin Lam Toh 1 *
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1 National Institute of Education, Nanyang Technological University, Nanyang Walk, SINGAPORE* Corresponding Author
Contemporary Mathematics and Science Education, 4(2), 2023, ep23019,
Published Online: 25 May 2023, Published: 01 July 2023
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Devising a plan is an important phase in the teaching and learning of mathematical problem-solving in a mathematics classroom. In this paper, we propose devise a plan (DP) model for scaffolding students in devising a plan to engage them in mathematical problem-solving for classroom instruction and beyond. Although mathematics educators have proposed problem-solving scaffold, mainly building on Polya’s (1945) and Schoenfeld’s (1985) problem-solving models, for authentic problem-solving in the classroom, the phase on devising a plan was generally brief. We expand on the scaffolding of the intermediate stages of devising the plan for teachers to teach problem-solving, with a more ambitious goal of enabling students to engage in independent problem-solving beyond the classrooms. Features that are used in the planning stage of problem-solving are identified through a systematic literature review. Our proposed DP model includes the use of both metacognitive strategies and problem-solving heuristics. The application of our proposed model was exemplified by the solution of three non-routine problem on proportionality.


Tay, Y. K., & Toh, T. L. (2023). A model for scaffolding mathematical problem-solving: From theory to practice. Contemporary Mathematics and Science Education, 4(2), ep23019.


  1. Anderson-Krathwohl, D. R., & Bloom, B. S. (2001). A taxonomy for learning, teaching, and assessing: A revision of Bloom’s taxonomy of educational objectives. Longman.
  2. Australian Education Council. (1990). A national statement on mathematics for Australian schools. Curriculum Corporation for Australian Education Council.
  3. Bos, R. (2017). Supporting problem solving through heuristic trees in an intelligent tutoring system. In G. Aldon, & J. Trgalova (Eds.), Proceedings of the 13th International Conference on Technology in Mathematics Teaching (pp. 436-439). Springer.
  4. Bos, R., & van den Bogaart, T. (2022). Heuristic trees as a digital tool to foster compression and decompression in problem-solving. Digital Experiences in Mathematics Education, 8(3), 157-182.
  5. Cockcroft Report. (1982). Mathematics counts. Her Majesty’s Stationary Office.
  6. Enright, B., & Beattie, J. (1989). Problem solving step by step in math. TEACHING Exceptional Children, 22(1), 58-59.
  7. Freeman-Green, S. M., O’Brien, C. L., Wood, C., & Hitt, S. B. (2015). Effects of the SOLVE strategy on the mathematical problem solving skills of secondary students with learning disabilities. Learning Disabilities Research & Practice, 30(2), 76-90.
  8. Jitendra, A. K., Hoff, K., & Beck, M. M. (1999). Teaching middle school students with learning disabilities to solve word problems using a schema-based approach. Remedial and Special Education, 20(1), 50-64.
  9. Jitendra, A. K., Star, J. R., Starosta, K., Leh, J. M., Sood, S., Caskie, G., Hughes, C. L., & Mack, T. R. (2009). Improving seventh grade students’ learning of ratio and proportion: The role of schema-based instruction. Contemporary Educational Psychology, 34(3), 250-264.
  10. Jung, E., Lim, R., & Kim, D. (2022). A schema-based instructional design model for self-paced learning environments. Education Sciences, 12(4), 271.
  11. Kahneman, D. (2011). Thinking, fast and slow. Farrar, Straus and Giroux.
  12. Lemmink, R. M. A. Z. (2019). Improving help-seeking behavior for online mathematical problem-solving lessons [Master’s thesis, Utrecht University].
  13. Leong, Y. H. (2009). Problems of teaching in a Singapore reform-oriented mathematics classroom. Lambert Academic Publishing.
  14. Leong, Y. H., Lu, P. C., & Toh, W. Y, K. (2021). Use of challenging items in instructional materials by Singapore secondary school mathematics teachers. In B. Kaur, & Y. H. Leong (Eds.), Mathematics instructional practices in Singapore secondary schools (pp. 231-248). Springer.
  15. Mason, J., Burton, L., & Stacey, K. (1982). Thinking mathematically. Addison-Wesley Publishing Company.
  16. NCTM. (1989). Principles and standards for school mathematics. National Council of Teachers of Mathematics.
  17. Polya, G. (1945). How to solve it. Princeton University Press.
  18. Popham, M., Adams, S., & Hodge, J. (2020). Self-regulated strategy development to teach mathematics problem solving. Intervention in School and Clinic, 55(3), 154-161.
  19. Reid, R., Lienemann, T. O., & Hagaman, J. L. (2014). Strategy instruction for students with learning disabilities. Guilford.
  20. Renkl, A., Atkinson, R. K., & Große, C. S. (2004). How fading worked solution steps works–A cognitive load perspective. Instructional Science, 32(1/2), 59-82.
  21. Roll, I., Baker, R. S. J. D., Aleven, V., & Koedinger, K. R. (2014). On the benefits of seeking (and avoiding) help in online problem-solving environments. Journal of the Learning Sciences, 23(4), 537-560.
  22. Rott, B., Specht, B., & Knipping, C. (2021). A descriptive phase model of problem-solving processes. ZDM Mathematics Education, 53(4), 737-752.
  23. Schoenfeld, A. H. (1985). Mathematical problem solving. Academic Press.
  24. Toh, T. L. (2011). Making mathematics practical: an approach to problem solving. World Scientific.
  25. Toh, T. L., Quek, K. S., & Tay, E. G. (2008a). Mathematical problem solving–A new paradigm. Connected maths: MAV yearbook, 356-365.
  26. Toh, T. L., Quek, K. S., & Tay, E. G. (2008b). Problem solving in the mathematics classroom (junior college). National Institute of Education, Nanyang Technological University.
  27. Wilson, J. W., Fernandez, M. L., & Hadaway, N. (1993). Mathematical problem solving. In P. S. Wilson (Ed.), Research ideas for the classroom: High school mathematics (pp. 57-77). MacMillan.