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Mathematical Problem Solving for Everyone: A Design Experiment

  • Quek, Khiok Seng (National Institute of Education, Nanyang Technological University) ;
  • Dindyal, Jaguthsing (National Institute of Education, Nanyang Technological University) ;
  • Toh, Tin Lam (National Institute of Education, Nanyang Technological University) ;
  • Leong, Yew Hoong (National Institute of Education, Nanyang Technological University) ;
  • Tay, Eng Guan (National Institute of Education, Nanyang Technological University)
  • Received : 2011.02.10
  • Accepted : 2011.03.25
  • Published : 2011.03.30

Abstract

An impetus for reviving research in mathematical problem solving is the recent advance in methodological thinking, namely, the design experiment ([Gorard, S. (2004). Combining methods in educational research. Maidenhead, England: Open University Press.]; [Schoenfeld, A. H. (2009). Bridging the cultures of educational research and design. Educational Designer. 1(2). http://www.educationaldesigner.orgied/volume1/issue21]). This methodological approach supports a "re-design" of contextual elements to fulfil the overarching objective of making mathematical problem solving available to all students of mathematics. In problem solving, components critical to successful design in one setting that may be adapted to suit another setting include curriculum design, assessment strategy, teacher capacity, and instructional resources. In this paper, we describe the implementation, over three years, of a problem solving module into the main mathematics curriculum of an Integrated Programme school in Singapore which had sufficient autonomy to tailor-fit curriculum to their students.

Keywords

References

  1. Armstrong, H. E. (1891). The teaching of scientific method. London: Macmillan.
  2. Black, P. (2009). In response to: Alan Schoenfeld. Educational Designer. 1(3).
  3. Black, P. & Wiliam, D. (1999). Assessment for Learning: Beyond the Black Box. Assessment Reform Group, University of Cambridge, School of Education.
  4. Branca, N. A. (1980). Problem solving as a goal, process, and basic skill. In: S. Krulik, & R. E. Reys (Eds.), Problem solving in school mathematics (pp. 3-8), 1980 Yearbook. Reston, VA: National Council of Teachers of Mathematics.
  5. Caldwell, J. & Ram, Y. M. (1999). Mathematical Modeling. Concepts and Case Studies. Kluwer.
  6. Curriculum Planning and Development Division. (2006). Mathematics Syllabus: Primary. Ministry of Education. Singapore.
  7. De Lange, J.; Keitel C.; Huntley, I. D. & Niss, M. (1993). Innovation in Maths Education by Modelling and Applications. Ellis Horwood, 1993.
  8. Dossey, J. A.; McCrone, S. S.; Giordano, F. R. & Weir, M. D. (2002). Mathematics methods and modeling methods for today's mathematics classroom: A contemporary approach to teaching grades 7-12. Pacific Grove, CA: Brooks/Cole. ME 2002e.04419
  9. English, L.; Lesh, R. & Fennewald, T. (2008). Future Directions and Perspectives for Problem Solving Research and Curriculum Development. Paper presented at the 11th International Conference on Mathematical Education, 6-13 July 2008 in Monterrey, Mexico.
  10. Fan, L. H. & Zhu, Y. (2000). Problem solving in Singaporean secondary mathematics textbooks. The Mathematics Educator 5(1/2), 117-141. ME 2001d.03653
  11. Fong, H. K. (1996). An integrated approach for teaching mathematical problem solving. The Mathematics Educator 1(1), 36-44. ME 1997e.02911
  12. Ginsburg, A.; Leinwand, S.; Anstrom, T.; Pollock, E. (2005). What the United States can learn from Singapore's world-class mathematics system (and what Singapore can learn from the United States): An exploratory study. American Institutes for Research. Washington, DC: AIR ERIC ED491632
  13. Gorard, S. (2004). Combining methods in educational research. Maidenhead, England: Open University Press.
  14. Halmos, P. (1980). The heart of mathematics. American Mathematical Monthly 87(7), 519-524. ME 1981f.01220 ERIC EJ232891 https://doi.org/10.2307/2321415
  15. Hammerness, K.; Jaramillo, R.; Unger, C. & Wilson, G. (1998). What do students in Teaching for Understanding classrooms understand? In: Stone Wiske (Ed.), Teaching for Understanding-Linking Research with Practice. San Francisco, CA: Jossey-Bass.
  16. Ho, K. F. & Hedberg, J. G. (2005). Teachers' pedagogies and their impact on students' mathematical problem solving. Journal of Mathematical Behavior 24(3-4), 238-252. ME 2007c.00147 https://doi.org/10.1016/j.jmathb.2005.09.006
  17. 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: L. C. J. Ee, B. Kaur, N. H. Lee & B. H. Yeap (Eds.), New' literacies': Educational response to a knowledge-based society (598-604). Singapore: Educational Research Association & Association of Mathematics Educators.
  18. Houston, S. K.; Blum, W.; Huntley, I. & Neill, N. T. (1997). Teaching and Learning Mathematical Modeling. Albion Publishing, Chichester. ME 1997d.02051
  19. Kilpatrick, J. (1985). A retrospective account of the past 25 years of research on teaching mathematical problem solving. In: E. A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 1-15). Hillsdale, NJ: Lawrence Erlbaum Associates.
  20. Krulik, S. (Ed.) (1980). Problem solving in school mathematics. Yearbook of the National Council of Teachers of Mathematics. Reston, VA: NCTM. ME 1982f.06611
  21. Krulik, S. & Rudnik, J. A. (1980). Problem solving: A handbook for teachers. Boston: Allyn & Bacon. ERIC ED287731
  22. Leong, Y. H.; Dindyal, J.; Tay, E. G.; Toh, T. L.; Quek, K. S. & Lou, S. T. (submitted to ZDM). Teacher preparation for a problem solving curriculum in Singapore.
  23. Lesh, R. & Zawojewski, J. (2007). Problem solving and modeling. In: F. Lester (Ed.), The second handbook of research on mathematics teaching and learning (pp. 763-804). Charlotte, NC: Information Age Publishing.
  24. Lovitt, C. (2000). Investigations as a central focus for a Mathematics curriculum. Retrieved April 20, 2010, from http://www.aamt.edu.au/Professional-learning/Professional-reading/Investigations-as-a-central-focus-for-a-Mathematics-curriculum. ERIC ED459082
  25. Mason, J. Burton, L. & Stacey, K. (1982). Thinking Mathematically. London: Addison-Wesley. ME 1984c.02646
  26. McLeod, D. B. (1992). Research on affect in mathematics education: a reconceptualization. In: D. A. Grouws (Ed.), NCTM Handbook of Research on Mathematics Teaching and Learning (pp. 575-596), New York, NY: Macmillan.
  27. Middleton, J.; Gorard, S.; Taylor, C. & Bannan-Ritland, B. (2006). The 'compleat' design experiment: from soup to nuts. Department of Educational Studies Research Paper 2006/05 University of York.
  28. Millar, R. (1991). A means to an end: The role of processes in science education. In: B. Woolnough (Ed.), Practical science (pp. 44-52). Buckingham, UK: Open University Press.
  29. National Council of Teachers of Mathematics. (1980). An agenda for action: Recommendations for school mathematics. Reston, VA: Author. ME 1980x.01275 ERIC ED186265
  30. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. ME 1989k.00892 ERIC ED304336
  31. National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: Author.
  32. National Institute of Education (2008). Research Framework. Singapore: National Institute of Education
  33. National Research Council (1996). National Science Education Standards. Washington, DC: National Academy of Sciences. ERIC ED391690
  34. Adams, L. M.; Tung, K. K.; Warfield, V. M.; Knaub, K.; Yong, D. & Mudavanhu, B. (2002). Middle school mathematics comparisons for Singapore Mathematics, Connected Mathematics Program, and Mathematics in Context (including comparisons with the NCTM Principles and Standards 2000). Report submitted to the National Science Foundation by the Department of Applied Mathematics, University of Washington. Retrieved from University of Washington, Department of Applied Mathematics website: http://www.amath.washington.edu/-adams/full.pdf
  35. Polya, G. (1945). How to solve it. Princeton: Princeton University Press. ME 1990j.01910
  36. Polya, G. (1961). Mathematical discovery. New York, NY: John Wiley & Sons.
  37. Polya, G. (1966). On teaching problem solving. In: E. G. Begle (Ed.), The role of axiomatics and problem solving in mathematics (pp. 123-129). Boston, MA: Ginn and Company.
  38. Polya, G. (1981). Mathematical Discovery: On Understanding, Learning and Teaching Problem Solving Combined Edition. New York: John Wiley. ME 1983e.01733
  39. Polya, G. (1990). Mathematics and Plausible Reasoning (Volume I): Induction and Analogy in Mathematics. Princeton: Princeton University Press. ME 1988f.00369
  40. Polya, G. (1990). Mathematics and Plausible Reasoning (Volume II): Patterns of Plausible Inference. Princeton: Princeton University Press. ME 1977a.00026
  41. Robitaille, D. & Dirks, M. (1982). Models for the mathematics curriculum. For the Learning of Mathematics 2(3), 3-21. ME 1983b.06921
  42. Schoenfeld, A. (1985). Mathematical problem solving. Orlando, FL: Academic Press. ME 1986a.01069
  43. 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.
  44. Schoenfeld, A. H. (1994). A discourse on methods. J. Res. Math. Educ. 25(6), 697-710. ME 1995d.02179 ERIC EJ498200 https://doi.org/10.2307/749581
  45. Schoenfeld, A. H. (2007). Bridging the cultures of educational research and design. Educational Designer. 1(2). http://www.educationaldesigner.org/ed/volume1/issue2/
  46. Schoenfeld, A. H. (2009). Problem solving in the United states, 1970-2008: Research and theory, practice and politics. ZDM: The International Journal on Mathematics Education 39, 537-551. ME 2009e.00288
  47. Shulman, L. S. & Shulman, J. H. (2004). How and what teachers learn: A shifting perspective. Journal of Curriculum Studies 36(2), 257-271. ERIC EJ695074 https://doi.org/10.1080/0022027032000148298
  48. Silver, E. A.; Ghousseini, H.; Gosen, D.; Charalambous, C. & Strawhun, B. T. F. (2025). Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom. Journal for Mathematical Behavior 24(3/4), 287-301. ME 2007c.00227 ERIC EJ803652 https://doi.org/10.1016/j.jmathb.2005.09.009
  49. Stacey, K. (1989). Finding and using patterns in linear generalising problems. Educational Studies in Mathematics 20, 147-164. ME 1990e.02959 ERIC EJ395740 https://doi.org/10.1007/BF00579460
  50. Stacey, K. (2005). The place of problem solving in contemporary mathematics curriculum documents. Journal of Mathematical Behavior 24(3/4), 341-350. ME 2007c.00172 ERIC EJ803666 https://doi.org/10.1016/j.jmathb.2005.09.004
  51. Stacey, K. & del Beato, C. (1990). Sources of Certainty and Uncertainty in Mathematical Problem Solving. Retrieved April 23, 2010, from http://staff.edfac.unimelb.edu.au/-kayecs/publications/1990to96/Staceydel%20Beato%20-%20Students.pdf.
  52. Stanic, G. M. A. & Kilpatrick, J. (1988). Historical perspectives on problem solving in the mathematics curriculum. In: R. I. Charles & E. A. Silver (Eds.), The teaching and assessing of problem solving (pp. 1-22). Reston, VA: National Council of Teachers of Mathematics.
  53. Stiggins, R. (2002). Assessment Crisis: The Absence of Assessment FOR Learning. Phi Delta Kappan, 83(10), 758-765. ERIC EJ648696 https://doi.org/10.1177/003172170208301010
  54. Tay, E. G.; Quek, K. S.; Toh, L.; Dong, F. & Ho, F. H. (2007). Mathematical problem solving for integrated programme students: The practical paradigm. Paper presented at the 4th East Asia Regional Conference on Mathematics Education.
  55. Toh, T. L.; Quek, K. S.; Leong, Y. H.; Dindyal, J. & Tay, E. G. (in press). Assessing problem solving in the mathematics curriculum: A new approach. In: B. Kaur & K. Y. Wong (Eds.), Association of Mathematics Educators Yearbook. Singapore: AME.
  56. Ullman, D. G. (1992). The Mechanical Design Process. New York: McGraw Hill.
  57. Wallas. G. (1926). The art of thought. New York: Harcourt, Brace.
  58. Woolnough, B. & Allsop, T. (1985). Practical work in science-Cambridge science education series. Cambridge, UK: Cambridge University Press.
  59. 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. 57-78). Reston, VA: National Council of Teachers of Mathematics.
  60. Yeap, B. H. & Kaur, B. (1998). Mathematical problem solving, thinking and creativity: Emerging themes for classroom instruction. The Mathematics Educator 3(2), 108-119.
  61. Yeo, S. M. & Zhu, Y. (2005). Higher-order thinking in Singapore mathematics classrooms. Proceedings of the International conference on education: Redesigning pedagogy - Research, policy and practice. Singapore: Centre for Research in Pedagogy and Practice, National Institute of Education.