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ZDM

, Volume 51, Issue 1, pp 151–168

# The why, what and how of the ‘Model’ method: a tool for representing and visualising relationships when solving whole number arithmetic word problems

• Berinderjeet Kaur
Original Article

## Abstract

The goal of this paper is to present a succinct consolidation of the why, what and how of the ‘Model’ method as there is a lack of such documentation for researchers, practitioners and policy makers thus far. The ‘Model’ method is a tool for representing and visualising relationships when solving whole number arithmetic (WNA) word problems. The paper traces the history of the method in the Singapore school mathematics curriculum, exemplifies the three basic models of the method, explores its efficacy and assesses its role as a problem solving heuristic in the Singapore primary school mathematics curriculum. The findings show that the method, a tool for representing and visualising relationships between quantities when solving whole number arithmetic word problems, was introduced in 1983 as part of the primary school mathematics curriculum for schools in Singapore. The method comprises three basic models, namely the part-whole model, the comparison model and the change model. Past studies carried out both in Singapore and other parts of the world, related to the use of the method and its impact on student learning show that the method does help students, including those with mathematical difficulties, improve in their ability to solve whole number arithmetic word problems. Perceptions of expert primary school mathematics teachers show that they engage students in solving whole number arithmetic word problems through Polya’s framework and also Newman’s strategies. They encourage students to represent on diagrams the givens and goals of a problem as this is a necessary step in visualising and recognising relationships. To do so they may use the method. Students may also use any other heuristic to solve the whole number arithmetic word problems.

## Keywords

Model method Whole number arithmetic Word problems Mathematical problem solving Primary school mathematics Singapore

## References

1. Ang, W. H. (2008). Singapore’s textbook experience 1965–97: Meeting the needs of curriculum change. In S. K. Lee, C. B. Goh, B. Fredriksen & J. P. Tan (Eds.), Toward a better future: Education and training for economic development in Singapore since 1965 (pp. 69–95). Washington, DC: World Bank.Google Scholar
2. Ang, W. H., & Yeoh, O. C. (1990). 25 years of curriculum development. In J. S. K. Yip & W. K. Sim (Eds.), Evolution of educational excellence—25 years of education in the Republic of Singapore (pp. 81–106). Singapore: Longman Singapore Publishers.Google Scholar
3. Bass, H. (2015). Quantities, numbers, number names, and the real number line. In X. H. Sun, B. Kaur, & J. Novotna (Eds.) Conference proceedings of ICMI study 23: Primary mathematics study on whole numbers (pp. 10–20). Macau, Macao SAR: University of Macau.Google Scholar
4. Bednarz, N., & Janvier, B. (1996). A problem-solving perspective on the introduction to algebra. In N. Bednarz, C. Kieran & L. Lee (Eds.), Approaches to algebra: Perspectives for learning and teaching (pp. 115–136). The Netherlands: Kluwer Academic Publishers.
5. Bruner, J. S. (1973). Beyond the information given: Studies in the psychology of knowing. Oxford: W.W. Norton.Google Scholar
6. Curriculum Development Institute of Singapore (CDIS). (1987). A report on primary mathematics project (July 1980–Dec. 1987). Singapore: Curriculum Development Institute of Singapore (CDIS).Google Scholar
7. Dienes, Z. P. (1971). The elements of mathematics. New York: Herder and Herder.Google Scholar
8. Ferrucci, B. J., Kaur, B., Carter, J. A., & Yeap, B. H. (2008). Using a model approach to enhance algebraic thinking in the elementary school mathematics classroom. In C. E. Greenes & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics (pp. 195–209). Reston: National Council of Teachers of Mathematics.Google Scholar
9. Foong, P. Y. (2010). Problem solving in mathematics. In P. Y. Lee & N. H. Lee (Eds.), Teaching primary school mathematics (2nd edn., pp. 54–79). Singapore: McGraw-Hill.Google Scholar
10. Goh, S. P. (2009). Primary 5 pupils’ difficulties in using the model method for solving complex relational word problems. Unpublished Master in Education dissertation. Singapore: National Institute of Education.Google Scholar
11. Jan, S., & Rodrigues, S. (2012). Model drawing strategy: A tool to link abstract words to real life. International Researcher, 1(4), 137–148.Google Scholar
12. Kaur, B. (2015). The model method—a tool for representing and visualising relationships. In X. H. Sun, B. Kaur, & J. Novotna (Eds.) Conference proceedings of ICMI study 23: Primary mathematics study on whole numbers (pp. 448–455). Macau, Macao SAR: University of Macau.Google Scholar
13. Kaur, B., Koay, P. L., & Yap, S. F. (2004). IPMA report. An international comparative study on the teaching and learning of mathematics in primary schools. Singapore: National Institute of Education.Google Scholar
14. Kho, T. H. (1987). Mathematical models for solving arithmetic problems. In Proceedings of the 4th Southeast Asian conference on mathematics education (ICMI-SEAMS) (pp. 345–351). Singapore.Google Scholar
15. Koleza, E. (2015). The bar model as a visual aid for developing complementary/variation problems. In K. Konrad & N. Vondrova (Eds.), Proceedings of CERME 9 (The ninth congress of the European Society for research in mathematics education), February 2015, Prague, Czech Republic.Google Scholar
16. Leh, J. (2011). Mathematics word problem solving: An investigation into schema-based instruction in a computer-mediated setting and a teacher-mediated setting with mathematically low-performing students. Doctoral Dissertation. Retrieved from ProQuest Dissertation and Theses database (MSTAR_851704601).Google Scholar
17. Mahoney, K. (2012). Effects of Singapore’s model method on elementary student problem-solving performance: Single-case research. Northeastern University (School of Education) Education Doctoral Theses. Paper 70. http://hdl.handle.net/2047/d20002962.
18. Mason, J. (2018). Structuring structural awareness: A commentary on Chap. 13. In M. G.B. Bussi & X. H. Sun (Eds.) Building the foundation: Whole numbers in the primary grades (pp. 325–340). Berlin: New ICMI Study Series, SpringerOpen.
19. Mellone, M., & Ramploud, A. (2015). Additive structure: An educational experience of cultural transposition. In X. H. Sun, B. Kaur, & J. Novotna (Eds.) Conference proceedings of ICMI study 23: Primary mathematics study on whole numbers (pp. 567–574). Macau, Macao SAR: University of Macau.Google Scholar
20. Ministry of Education. (1979). Report on the Ministry of Education 1978 (Prepared by Dr Goh Keng Swee and the Education Study Team). Singapore: Ministry of Education.Google Scholar
21. Ministry of Education. (1981). Diagnostic tests on the basic skills of mathematics for primary school pupils. Singapore: Ministry of Education.Google Scholar
22. Ministry of Education. (2009). The Singapore model method for learning mathematics. Singapore: Ministry of Education.Google Scholar
23. Morin, L. L., Watson, S. M. R., Hester, P., & Raver, S. (2017). The use of a bar model drawing to teach word problem solving to students with mathematics difficulties. Learning Disability Quarterly, 40(2), 91–104.
24. Murata, A. (2008). Mathematics teaching and learning as a mediating process: The case of tape diagrams. Mathematical Thinking and Learning, 10(4), 374–406. .
25. Newman, A. (1983). Strategies for diagnosis and remediation. Melbourne: Harcourt Brace Jovanovich Group.Google Scholar
26. 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
27. Ng, S. F., & Lee, K. (2009). The model method: Singapore children’s tool for representing and solving algebraic word problems. Journal for Research in Mathematics Education, 40(3), 282–313.Google Scholar
28. Poh, B. K. (2007). Model method: Primary three pupils’ ability to use models for representing and solving word problems. Unpublished Master in Education dissertation. National Institute of Education, Singapore.Google Scholar
29. Polya, G. (1957). How to solve it. Princeton: Princeton University Press.Google Scholar
30. Powell, S. R. (2011). Solving word problems using schemas: A review of the literature. Learning Disabilities Research and Practice, 26(2), 94–108.
31. Resnick, L. B., Bill, V. L., Lesgold, S. B., & Leer, N. M. (1991). Thinking in arithmetic class. In B. Means, C. Chelemer & M. S. Knapp (Eds.), Teaching advanced skills to at-risk students (pp. 27–53). San Francisco: Jossey-Bass.Google Scholar
32. Schmittau, J. (2005). The development of algebraic thinking—a Vygotskian perspective. ZDM, 37(1), 16–22.Google Scholar
33. Schmittau, J. (2011). The role of theoretical analysis in developing algebraic thinking: A Vygotskian perspective. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 71–86). Berlin: Springer.
34. Schmittau, J., & Morris, A. (2004). The development of algebra in the elementary mathematics curriculum of V. V. Davydov. The Mathematics Educator, 8(1), 60–87.Google Scholar
35. Sun, X. H., & Bussi, M. G. B. (2018). Language and cultural issues in the teaching and learning of WNA. In M. G. B. Bussi & X. H. Sun (Eds.), Building the foundation: Whole numbers in the primary grades (pp. 35–70). Berlin: New ICMI Study Series, Springer.
36. Yip, J. S. K., & Sim, W. K. (Eds.). (1990). Evolution of educational excellence—25 years of education in the Republic of Singapore (pp. 81–106). Singapore: Longman Singapore Publishers.Google Scholar

## Copyright information

© FIZ Karlsruhe 2018

## Authors and Affiliations

• Berinderjeet Kaur
• 1
1. 1.National Institute of EducationSingaporeSingapore