Mathematics Education in Singapore pp 141-164 | Cite as

# Problem Solving in the Singapore School Mathematics Curriculum

## Abstract

Problem solving has been the heart of the Singapore school mathematics curriculum since the early 1990s after being adopted as the goal of school mathematics education. Since its adoption, it has captured the interest of many Singapore educators and researchers. It appears that problem solving will continue to be a very active research area since there is great interest in the very high level of performance of Singapore students in international comparative studies such as TIMSS and PISA. This chapter begins with a re-categorization of the research work done to date on problem solving in Singapore using the Singapore mathematics curriculum framework by integrating two classifications done by Foong in 2009 and Chan in 2014, respectively, and including work done since 2011 that was not reported in either survey. The earlier research focused on addressing the readiness of students for mathematical problem solving (MPS) from the perspective of the Singapore mathematics curriculum framework; the later research tended to emphasize the enactment of MPS in the Singapore mathematics classroom and teacher education. This chapter gives more detail to this later research with an emphasis on the enactment of Pólya’s stages in solving structured problems.

## Keyword

Mathematical problem solving Pólya’s model Pre-service teacher education Real-world context## References

- Ahmad Ibrahim, E. (2004). Computer-supported collaborative problem solving and anchored instruction in a mathematics classroom: An exploratory study.
*International Journal of Learning and Technology,**1*(1), 16–39.Google Scholar - Brown, A. L. (1992). Design experiments: Theoretical and methodological challenges in creating complex interventions.
*The Journal of the Learning Sciences,**2,*137–178.Google Scholar - Chan, C. M. E. (2005). Engaging students in open-ended mathematics problem-tasks: A sharing on teachers’ production and classroom experience. Paper presented at ICMI-EARCOME 3 Conference, Shanghai.Google Scholar
- Chan, C. M. E. (2010). Tracing primary 6 pupils’ model development within the mathematical modelling process.
*Journal of Mathematical Modelling and Application,**1*(3), 40–57.Google Scholar - Chan, C. M. E. (2011). Primary 6 students’ attitudes towards mathematical problem-solving in a problem-based learning setting.
*The Mathematics Educator,**13*(1), 15–31.Google Scholar - Chan, C. M. E. (2015). A review of mathematical problem solving research involving students in Singapore mathematics classrooms (2001 to 2011): What’s done and what more can be done. In B. Sriraman, J. Cai, K.-H. Lee, L. H. Fan, Y. Shimuzu, C. S. Lim, & K. Subramanium (Eds.),
*The first sourcebook on Asian research in mathematics education: China, Korea, Singapore, Japan, Malaysia and India*(pp. 233–257). Charlotte, NC: Information Age Publishing.Google Scholar - Chang, S. H. (2004).
*Sense-making in solving arithmetic word problems among Singapore primary school students*(Unpublished master’s dissertation). National Institute of Education, Singapore.Google Scholar - Chang, C. Y. (2005).
*An open-ended approach to promote higher order thinking in mathematics among Secondary Two Express students*(Unpublished master’s dissertation). National Institute of Education, Singapore.Google Scholar - Chow, I. V. P. (2004).
*Impact of open-ended problem solving as an alternative assessment on Secondary One Mathematics students*(Unpublished Master’s dissertation). National Institute of Education, Singapore.Google Scholar - Chua, P. H., & Fan, L. H. (2007, June).
*Mathematical problem posing characteristics of Secondary 3 Express students in Singapore.*Paper presented at the fourth East Asia Regional Conference on Mathematics Education Conference, Universiti Sains Malaysia.Google Scholar - Collins, A. (1999). The changing infrastructure of education research. In E. C. Langemann & L. S. Shulman (Eds.),
*Issues in education research*(pp. 15–22). San Francisco, CA: Jossey-Bass.Google Scholar - Dindyal, J., Quek, K. S., Leong, Y. H., Toh, T. L., Tay, E. G., & Lou, S. T. (2010, July). Problems for a problem solving curriculum. In L. Sparrow, B. Kissane, & C. Hurst (Eds.),
*Shaping the future of mathematics education*(Vol. 2, pp. 749–752).Google Scholar - Dindyal, J., Tay, E. G., Toh, T. L., Leong, Y. H., & Quek, K. S. (2012). Mathematical problem solving for everyone: A new beginning.
*The Mathematics Educator,**13,*51–70.Google Scholar - Dindyal, J., Toh, T. L., Quek, K. S., Leong, Y. H., & Tay, E. G. (2009, June).
*Devising a problem solving curriculum*. Paper presented at the Redesigning Pedagogy International Conference 2009, Singapore.Google Scholar - Dong, F. M., Lee, T. Y., Tay, E. G., & Toh, T. L. (2002). Performance of Singapore Junior College students on some nonroutine problem. In D. Edge & Y. B. Har (Eds.),
*EARCOME, 2002*(pp. 71–77). Singapore: NA.Google Scholar - Fan, L., & Zhu, Y. (2007). From convergence to divergence: The development of mathematical problem solving research, curriculum, and classroom practice in Singapore.
*ZDM Mathematics Education,**39*(5–6), 491–501.Google Scholar - Fan, L., & Zhu, Y. (2008). Using assessment performance in secondary school mathematics: An empirical study in a Singapore classroom.
*Journal of Mathematics Education,**1*(1), 132–152.Google Scholar - Foong, P. Y. (2002). The role of problems to enhance pedagogical practices in the Singapore mathematics classroom.
*The Mathematics Educator,**6*(2), 15–31.Google Scholar - Foong, P. Y. (2005). Developing creativity in the Singapore mathematics classroom.
*Thinking Classroom,**6*(4), 14–20.Google Scholar - Foong, P. Y. (2009). Review of research on mathematical problem solving in Singapore. In K. Y. Wong, P. Y. Lee, B. Kaur, P. Y. Foong, & S. F. Ng (Eds.),
*Mathematics education: The Singapore journey*(pp. 263–300). Singapore: World Scientific.Google Scholar - Foong, P. Y., Yap, S. F., & Koay, P. L. (1996). Teachers’ concerns about the revised mathematics curriculum.
*The Mathematics Educator,**1*(1), 99–110.Google Scholar - Goh, S. P (2009).
*Primary 5 students’ difficulties in using the model method for solving complex relational word problems*(Unpublished master’s dissertation). National Institute of Education, Singapore.Google Scholar - Gorard, S. (2004).
*Combining methods in educational research*. Maidenhead, England: Open University Press.Google Scholar - Hedberg, J. G., Wong, K. Y., Ho, K. F., Lioe, L., & Tiong, J. (2005).
*Developing the repertoire of heuristics for mathematical problem solving*. Executive Summary Report for Project No. CRP 38/03 TSK. Available at http://repository.nie.edu.sg/jspui/bitstream/10497/258/4/CRP38_03TSK_Summary.pdf. Accessed on June 25, 2011. - Heng, C. H. J. (2007).
*Primary pupils’ ability to engage in sense making when solving word problems*(Unpublished master’s dissertation). National Institute of Education, Singapore.Google Scholar - Ho, G. L. (2007). A cooperative learning programme to enhance mathematical problem solving.
*The Mathematics Educator,**10*(1), 59–80.Google Scholar - Ho, K. F., & Hedberg, J. G. (2005). Teachers’ pedagogies and their impact on students’ mathematical problem solving.
*Journal of Mathematical Behaviour,**24*(3 & 4), 238–252.Google Scholar - Ho, S. Y., Lee, S., & Yeap, B. H. (2001). Children posing word problems during a paper-and-pencil test: Relationship between achievement and problem posing ability. In J. Ee, B. Kaur, N. H. Lee, & B. H. Yeap (Eds.),
*New “Literacies”: Educational responses to a knowledge-based society*(pp. 598–604). Singapore: ERA.Google Scholar - Ho, W. K., Yap, R. A. S., Tay, E. G., Leong, Y. H., Toh, T. L., Quek, K. S., et al. (in-press). Understanding the sustainability of a teaching innovation for problem solving: A systems approach. In P. Liljedahl (Ed.),
*Mathematical problem solving: Current themes, trends and research*(pp. 1–19). Burnaby: Springer.Google Scholar - Hung, D. W. L. (2001). Conjectured ideas as mediating artifacts for the appropriation of mathematical ideas.
*Journal of Mathematical Behaviour,**20,*247–262.Google Scholar - Kaur, B. (2009). Performance of Singapore students in trends in international mathematics and science studies (TIMSS). In K. Y. Wong, P. Y. Lee, B. Kaur, P. Y. Foong, & S. F. Ng (Eds.),
*Mathematics education: The Singapore journey*(pp. 439–463). Singapore: World Scientific.Google Scholar - Kaur, B., & Toh, T. L. (2011). Mathematical problem solving—Linking theory and practice. In O. Zaslavky & P. Sullivan (Eds.),
*Constructing knowledge for teaching secondary mathematics: Tasks to enhance prospective and practicing teacher learning*(pp. 177–188). New York: Springer.Google Scholar - Lee, C. M. (2002).
*Integrating the computer and thinking into the primary mathematics classroom*(Unpublished master’s dissertation). National Institute of Education, Singapore.Google Scholar - Lee, N. H. (2008).
*Enhancing mathematical learning and achievement of secondary one Normal (Academic) students using metacognitive strategies*(Unpublished doctoral dissertation). National Institute of Education, Singapore.Google Scholar - Lee, N. H. (2014). A metacognitive-based instruction for primary four students to approach non-routine mathematical word problems.
*ZDM Mathematics Education,**46,*465–480.Google Scholar - Leong, Y. H., Quek, K. S., Toh, T. L., Dindyal, J., & Tay, E. G. (2009, June).
*Teacher preparation for the problem solving curriculum*. Paper presented at the Redesigning Pedagogy International Conference, Singapore.Google Scholar - Leong, Y. H., Tay, E. G., Toh, T. L., Quek, K. S., Toh, P. C., & Dindyal, J. (2016a). Infusing mathematical problem solving in the mathematics curriculum: Replacement Units. In P. Felmer, E. Perhkonen, & J. Kilpatrick (Eds.),
*Posing and solving mathematical problems: Advances and new perspectives*(pp. 309–326). Geneva: Springer.Google Scholar - Leong, Y. H., Tay, E. G., Toh, T. L., Yap, R. A. S, Toh, P. C., Quek, K. S., & Dindyal, J. (2016b). Boundary objects within a replacement unit strategy for mathematics teacher development. In B. Kaur, O. N. Kwon, & Y. H. Leong (Eds.),
*Professional development of mathematics teachers: An Asian perspective*(pp. 189–208). Singapore: Springer.Google Scholar - Leong, Y. H., Toh, T. L., Quek, K. S., Dindyal, J., & Tay, E. G. (2010).
*Enacting a problem solving curriculum*. In L. Sparrow, B. Kissane, & C. Hurst (Eds.),*The mathematics education research group of Australasia: Shaping the future of mathematics education*(Vol. 2, pp. 745–748).Google Scholar - Leong, Y. H., Yap, S. F., Quek, K. S., Tay, E. G., & Tong, C. L. (2013). Encouraging problem-solving disposition in a Singapore classroom.
*International Journal of Mathematical Education in Science and Technology,**44*(8), 1257–1273.Google Scholar - Leong, Y. H., Yap, R. A., Toh, T. L., Tay, E. G., Quek, K. S., Toh, P. C., et al. (in-press). Students’ perceptions about an undergraduate mathematics problem solving course. In M. Stein (Ed.),
*A life’s time for mathematics education and problem solving.*Google Scholar - Lester, F. K. (1994). Musing about mathematical problem-solving research: 1970–1994.
*Journal of Research in Mathematics Education,**25,*660–676.Google Scholar - Lioe, L. T., Ho, K. F., & Hedberg, J. (2006). Students’ metacognitive problem solving strategies in solving open-ended problems in pairs. In W. D. Bokhorst-Heng, M. D. Osborne, & K. Lee (Eds.),
*Redesigning pedagogy: Reflection on theory and praxis*(pp. 243–260). Rotterdam, The Netherlands: Sense Publishers.Google Scholar - Looi, C. K., & Lim, K. S. (2009). From bar diagrams to letter-symbolic algebra: A technology-enabled bridging.
*Journal of Computer Assisted learning,**25*(4), 358–374.Google Scholar - McTighe, J., & Wiggins, G. (2004).
*Understanding by design: Professional development workbook*. Alexandria, VA: Association for Supervision and Curriculum Development.Google Scholar - Ministry of Education. (1990).
*Mathematics syllabus (lower secondary).*Singapore: Author.Google Scholar - Ministry of Education. (2006).
*Secondary mathematics syllabus*. Singapore: Author.Google Scholar - Ng, H. C. (2003).
*Benefits of using investigative tasks in the primary classroom*(Unpublished master’s thesis). National Institute of Education, Nanyang Technological University, Singapore.Google Scholar - Ng, W. L. (2006). Effects of an ancient Chinese mathematics enrichment programme on secondary school students achievements in mathematics.
*International Journal of Science and Mathematical Education,**4,*485–511.Google Scholar - Ng, K. E. D. (2010). Collective reasoning and sense making processes during a real-world mathematical project. In Y. Shimizu, Y. Sekiguchi, & K. Hino (Eds.), In search of excellence in mathematics education:
*Proceedings of the 5th East Asia Regional Conference on Mathematics Education*(Vol. 2, pp. 771–778). Tokyo, Japan: Japan Society of Mathematical Education.Google Scholar - Ng, S. F., & Lee, K. (2005). How primary five pupils use the model method to solve word problems.
*The Mathematics Educator,**9*(1), 60–83.Google Scholar - Poh, B. K. (2007).
*Model method: Primary three pupils’ ability to use models for representing and solving word problems*(Unpublished master’s dissertation). National Institute of Education, Singapore.Google Scholar - Pólya, G. (1954).
*How to solve it*. Princeton: Princeton University Press.Google Scholar - Quek, K. S. (2002).
*Cognitive characteristics and contextual influences in mathematical problem posing*(Unpublished Ph.D. dissertation). National Institute of Education, Nanyang Technological University, Singapore.Google Scholar - Quek, K. S., Toh, T. L., Dindyal, J., Leong, Y. H., Tay, E. G., & Lou, S. T. (2010, July). Resources for teaching problem solving: A problem to discuss. In L. Sparrow, B. Kissane, & C. Hurst (Eds.),
*Shaping the future of mathematics education*(Vol. 2, pp. 753–756).Google Scholar - Quek, K. S., Toh, T. L., Leong, Y. H., Dindyal, J., & Tay, E. G. (2009, June).
*Assessment in the problem solving curriculum*. Paper presented at the Redesigning Pedagogy International Conference 2009, Singapore.Google Scholar - Schoenfeld, A. H. (1985).
*Mathematical problem solving*. Orlando, FL: Academic Press.Google Scholar - Schoenfeld, A. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 334–370). New York: Macmillan.Google Scholar - Schoenfeld, A. H. (2007). Problem solving in the United States, 1970–2008: Research and theory, practice and politics.
*ZDM Mathematics Education,**39,*537–551.Google Scholar - Seoh, B. H. (2002).
*An open-ended approach to enhance critical thinking skill in mathematics among secondary five normal (Academic) pupils*(Unpublished master’s dissertation). National Institute of Education, Singapore.Google Scholar - Shroeder, T., & Lester, F. (1989). Developing understanding in mathematics via problem solving. In P. Traffon & A. Shulte (Eds.),
*New directions for elementary school mathematics: 1989 Yearbook*(pp. 31–42). Reston, VA: NCTM.Google Scholar - Tan, T. L. S. (2002).
*Using project work as a motivating factor in lower secondary mathematics*(Unpublished master’s thesis). National Institute of Education, Nanyang Technological University, Singapore.Google Scholar - Tay, E. G., Toh, T. L., Ho, F. H., Toh, P. C., Leong, Y. H., Quek, K. S., et al. (2016, July).
*Infusing mathematical problem solving into the mathematics curriculum: Feedback from teachers*. Paper presented at 13th International Congress on Mathematical Education, Hamburg, Germany.Google Scholar - Teo, A. L. (2005).
*Effects of an intervention programme on the sense-making ability of primary three pupils*(Unpublished master’s dissertation). National Institute of Education, Singapore.Google Scholar - Teo, O. M. (2006).
*A small-scale study on the effects of metacognition and beliefs on students in A-level sequences and series problem-solving*(Unpublished master’s dissertation). National Institute of Education, Singapore.Google Scholar - Teong, S. K. (2003). Metacognitive intervention strategy and word problem solving in a cognitive-apprenticeship-computer-based environment. In
*Proceedings of the Association for Active Educational Researchers Conference*, Auckland, New Zealand.Google Scholar - Teong, S. K., Hedberg, J. G., Ho, K. F., Lioe, L. T., Tiong, Y. S. J., Wong, K. Y., & Fang, Y. P. (2009). Developing the repertoire of heuristics for mathematical problem solving: Project 1. Final Technical Report for Project CRP1/04 JH. Singapore: Centre for Research in Pedagogy and Practice, National Institute of Education, Nanyang Technological University. http://hdl.handle.net/10497/4151.
- Toh, T. L. (2009). Arousing students’ curiosity and mathematical problem solving. In B. Kaur & B. H. Yeap (Eds.),
*AME Yearbook 2008*(pp. 241–262). Singapore: World Scientific.Google Scholar - Toh, T. L. (2010). Making decisions with mathematics: From mathematical problem solving to modelling. In B. Kaur & J. Dindyal (Eds.),
*Mathematical applications and modelling: AME Yearbook 2010*(pp. 1–18). Singapore: World Scientific.Google Scholar - Toh, T. L., Quek, K. S., Leong, Y. H., Dindyal, J., & Tay, E. G. (2009, July).
*Assessment in a problem solving curriculum*. In R. Hunter, B. Bicknell, & T. Burgess (Eds.),*MERGA 32 Conference Proceedings*(Vol. 2, pp. 686–690).Google Scholar - Toh, T. L., Quek, K. S., Leong, Y. H., Dindyal, J., & Tay, E. G. (2011a).
*Making mathematics practical: An approach to problem solving*. Singapore: World Scientific.Google Scholar - Toh, T. L., Quek, K. S., Leong, Y. H., Dindyal, J., & Tay, E. G. (2011b). Assessing problem solving in the mathematics curriculum: A new approach. In K. Y. Wong & B. Kaur (Eds.),
*AME Yearbook 2011: Assessment*(pp. 1–35). Singapore: World Scientific.Google Scholar - Toh, T. L., Quek, K. S., & Tay, E. G. (2008). Mathematical problem solving—A new paradigm. In J. Vincent, R. Pierce, & J. Dowsey (Eds.),
*Connected maths: MAV Yearbook 2008*(pp. 356–365). Melbourne: The Mathematical Association of Victoria.Google Scholar - Toh, T. L., Quek, K. S., Tay, E. G., Leong, Y. H., Toh, P. C., Ho, F. H., & Dindyal, J. (2013). Infusing problem solving into mathematics content course for pre-service secondary school mathematics teachers.
*The Mathematics Educator, 15*(1), 98–120.Google Scholar - Toh, P. C., Leong Y. H., Toh, T. L., Dindyal, J., Quek, K. S., Tay, E. G., & Ho, F. H. (2014). The problem-solving approach in the teaching of number theory.
*International Journal of Mathematical Education in Science and Technology, 45*(2), 241–255.Google Scholar - Wong, S. O. (2002).
*Effects of heuristics instruction on pupils’ achievement in solving non-routine problems*(Unpublished master’s dissertation). National Institute of Education, Singapore.Google Scholar - Wong, K. Y. (2007). Metacognitive awareness of problem solving among primary and secondary school students. In
*Proceedings of the Redesigning Pedagogy: Culture, Knowledge and Understanding Conference, Singapore.*Google Scholar - Wong, K. Y. (2008). Developing the repertoire of heuristics for mathematical problem solving. In M. Goos, R. Brown, & K. Makar (Eds.),
*Proceedings of the 31st Annual Conference of the Mathematics Education Research Group of Australasia*(pp. 589–595). Brisbane: MERGA.Google Scholar - Wong, S. O., & Lim-Teo, S. K. (2002). Effects of heuristics instruction on pupils’ mathematical problem-solving processes. In D. Edge & B. H. Yeap (Eds.),
*Proceedings of Second East Asia Regional Conference on Mathematics Education and Ninth Southeast Asian Conference on Mathematics Education Volume 2 Selected Papers*(pp. 180–186). Singapore: Association of Mathematics Educators.Google Scholar - Wood, T., & Berry, B. (2003). What does “design research” offer mathematics education?
*Journal of Mathematics Teacher Education,**6,*195–199.Google Scholar - Yeap, B. H., & Lee, N. H. (2002). Writing word problems as a learning tool: An exploratory study. In D. Edge & B. H. Yeap (Eds.),
*Proceedings of the Second East Asia Regional Conference on Mathematics Education & Ninth Southeast Asian Conference on Mathematics Education*(pp. 187–193). Singapore: Association of Mathematics Educators.Google Scholar - Yeo, K. K. J. (2005). Anxiety and performance on mathematical problem solving of secondary two students in Singapore.
*The Mathematics Educator,**8*(2), 71–83.Google Scholar - Yeo, K. K. J. (2006). Mathematical problem-solving heuristics used by secondary 2 students.
*The Korean Journal of Thinking & Problem Solving,**16*(2), 53–69.Google Scholar - Yeo, K. K. J. (2011). An exploratory study of primary 2 pupils approach to solve word problems.
*Journal of Mathematics Education,**4*(1), 19–30.Google Scholar