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“Looking Back” to Solve Differently: Familiarity, Fluency, and Flexibility

  • Hartono TjoeEmail author
Chapter
Part of the ICME-13 Monographs book series (ICME13Mo)

Abstract

The present study focuses on a specific step of Pólya’s problem-solving process, namely, “looking back” to solve a problem differently. In particular, it examines the extent to which the practice of “looking back” to solve differently has been integrated into mathematics classroom instruction in the United States. The findings of the present study indicate, to a certain degree, that this practice is little known to high school students, even those from some very selective schools. Moreover, it demonstrates that a high level of mathematical preparation might not be a sufficient condition for a student’s inclination to search for (let alone consider the need for) more than one solution method. Pedagogical implications of emphasizing the importance of connections between different mathematical topics are discussed.

References

  1. Bodemer, D., Plötzner, R., Feuerlein, I., & Spada, H. (2004). The active integration of information during learning with dynamic and interactive visualizations. Learning and Instruction, 14, 325–341.CrossRefGoogle Scholar
  2. Borwein, P., Liljedahl, P., & Zhai, H. (2014). Mathematicians on creativity. Washington, D.C.: Mathematical Association of America.Google Scholar
  3. Collins, A. (1991). Cognitive apprenticeship and instructional technology. In L. Idol & B. F. Jones (Eds.), Educational values and cognitive instruction: Implication for reform (pp. 121–138). Hillsdale, NJ: Erlbaum.Google Scholar
  4. Collins, A., Brown, J. S., & Newman, S. E. (1989). Cognitive apprenticeship. Teaching the crafts of reading, writing, and mathematics. In L. B. Resnick (Ed.), Knowing, learning, and instruction (pp. 453–493). Hillsdale, NJ: Erlbaum.Google Scholar
  5. Common Core State Standards Initiative. (2010). Common core state standards for mathematics. Retrieved from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf.
  6. Davis, P. J., & Hersh, R. (1981). The mathematical experience. Boston, MA: Birkhauser.Google Scholar
  7. de Jong, T., Ainsworth, S., Dobson, M., van der Hulst, A., Levonen, J., Reimann, P., et al. (1998). Acquiring knowledge in science and mathematics: The use of multiple representations in technology-based learning environments. In M. van Someren, P. Reimann, H. Boshuizen, & T. de Jong (Eds.), Learning with multiple representations (pp. 9–41). Oxford, England: Elsevier Sciences.Google Scholar
  8. Douady, R., & Perrin-Glorian, M.-J. (1989). Un processus d’apprentissage du concept d’aire de surface plane. Educational Studies in Mathematics, 20(4), 387–423.CrossRefGoogle Scholar
  9. Duncker, K. (1945). On problem-solving (L. S. Lees, Trans.). Psychological Monographs, 58(5), i-113 (Whole No. 270).CrossRefGoogle Scholar
  10. Eisenhart, M., Borko, H., Underhill, R., Brown, C., Jones, D., & Agard, P. (1993). Conceptual knowledge falls through the cracks: Complexities of learning to teach mathematics for understanding. Journal for Research in Mathematics Education, 24(1), 8–40.CrossRefGoogle Scholar
  11. Felmer, P., Pehkonen, E., & Kilpatrick, J. (2016). Posing and solving mathematical problems: Advances and new perspectives. New York, NY: Springer.CrossRefGoogle Scholar
  12. Garofalo, J., & Lester, F. K. (1985). Metacognition, cognitive monitoring, and mathematical performance. Journal for Research in Mathematics Education, 16(3), 163–176.CrossRefGoogle Scholar
  13. Gick, M. L., & Holyoak, K. J. (1980). Analogical problem solving. Cognitive Psychology, 12, 306–355.CrossRefGoogle Scholar
  14. Große, C. S., & Renkl, A. (2006). Effects of multiple solution methods in mathematics learning. Learning and Instruction, 16(2), 122–138.CrossRefGoogle Scholar
  15. Hadamard, J. (1945). The psychology of invention in the mathematical field. Princeton, NJ: Princeton University Press.Google Scholar
  16. Hersant, M. (2011). Correspondance entre élèves: conditions d’une activité mathématique « créative » et problématisée à la fin du lycée. Educational Studies in Mathematics, 78(3), 343–370.CrossRefGoogle Scholar
  17. Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Hillsdale, NJ: Erlbaum.Google Scholar
  18. Hogan, D. M., & Tudge, J. R. (1999). Implications of Vygotsky’s theory for peer learning. In A. M. O’Donnell & A. King (Eds.), Cognitive perspectives on peer learning (pp. 39–65). Mahwah, NJ: Erlbaum.Google Scholar
  19. Kaiser, G., & Schwarz, B. (2006). Mathematical modelling as bridge between school and university. Zentralblatt für Didaktik der Mathematik, 38(2), 196–208.CrossRefGoogle Scholar
  20. 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: Erlbaum.Google Scholar
  21. Leikin, R. (2009). Exploring mathematical creativity using multiple solution tasks. In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativity in mathematics and the education of gifted students (pp. 129–145). Rotterdam, The Netherlands: Sense Publisher.Google Scholar
  22. Leikin, R., & Lev, M. (2007). Multiple solution tasks as a magnifying glass for observation of mathematical creativity. In J. H. Woo, H. C. Lew, K. S. Park, & D. Y. Seo (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 161–168). Seoul, Korea: PME.Google Scholar
  23. Leikin, R., & Levav-Waynberg, A. (2007). Exploring mathematics teacher knowledge to explain the gap between theory-based recommendations and school practice in the use of connecting tasks. Educational Studies in Mathematics, 66(3), 349–371.CrossRefGoogle Scholar
  24. Lesh, R. (1985). Conceptual analyses of problem solving performance. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 309–330). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
  25. Lester, F. K. (1994). Musings about mathematical problem-solving research: The first 25 years in the JRME. Journal for Research in Mathematics Education, 25(6), 660–675.CrossRefGoogle Scholar
  26. Liljedahl, P. (2016). Building thinking classrooms: Conditions for problem-solving. In P. Felmer, E. Pehkonen, & J. Kilpatrick (Eds.), Posing and solving mathematical problems. New York, NY: Springer.CrossRefGoogle Scholar
  27. Liljedahl, P., & Sriraman, B. (2006). Musings on mathematical creativity. For the Learning of Mathematics, 26(1), 20–23.Google Scholar
  28. Mason, J., Burton, L., & Stacey, K. (1982). Thinking mathematically. New York, NY: Addison Wesley.Google Scholar
  29. Mathematical Association of America. (2011). Retrieved from https://www.maa.org/math-competitions/about-amc.
  30. Michener, E. R. (1978). Understanding understanding mathematics. Cognitive Science, 2, 361–383.CrossRefGoogle Scholar
  31. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.Google Scholar
  32. Owen, E., & Sweller, J. (1985). What do students learn while solving mathematics problems? Journal of Educational Psychology, 77, 272–284.CrossRefGoogle Scholar
  33. Poincare, H. (1946). The foundations of science (G. B. Halsted, Trans.). Lancaster, PA: Science Press.Google Scholar
  34. Pólya, G. (1945). How to solve it. Princeton, NJ: Princeton University Press.Google Scholar
  35. Pressley, M., Forrest-Pressley, D. L., Elliott-Faust, D., & Miller, G. E. (1985). Children’s use of cognitive strategies, how to teach strategies, and what to do if they can’t be taught. In M. Pressley & C. J. Brainerd (Eds.), Cognitive processes in memory development. New York, NY: Springer.Google Scholar
  36. Reeves, L. M., & Weisberg, R. W. (1994). The role of content and abstract information in analogical transfer. Psychological Bulletin, 115, 381–400.CrossRefGoogle Scholar
  37. Resnick, L. B. (1989). Knowing, learning, and instruction. Hillsdale, NJ: Erlbaum.Google Scholar
  38. Rittle-Johnson, B., & Siegler, R. S. (1998). The relation between conceptual and procedural knowledge in learning mathematics: A review. In C. Donlan (Ed.), The development of mathematical skills (pp. 75–110). East Sussex, England: Psychology Press.Google Scholar
  39. Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561–574.CrossRefGoogle Scholar
  40. Santos-Trigo, M. (1996). An exploration of strategies used by students to solve problems with multiple ways of solution. Journal of Mathematical Behavior, 15, 263–284.CrossRefGoogle Scholar
  41. Santos-Trigo, M. (1998). Can routine problems be transformed into non-routine problems? Teaching Mathematics and Its Applications, 17(3), 132–135.CrossRefGoogle Scholar
  42. Schoenfeld, A. H. (1979a). Can heuristics be taught? In J. Lochhead & J. Clement (Eds.), Cognitive process instruction (pp. 315–338). Philadelphia, PA: Franklin Institute.Google Scholar
  43. Schoenfeld, A. H. (1979b). Explicit heuristic training as a variable in problem solving performance. Journal for Research in Mathematics Education, 10, 173–187.CrossRefGoogle Scholar
  44. Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic Press.Google Scholar
  45. Schoenfeld, A. H. (2008). Problem solving in the United States, 1970–2008: Research and theory, practice and politics. ZDM The International Journal of Mathematics Education, 39(5–6), 537–551.Google Scholar
  46. Silver, E. A. (1985). Research on teaching mathematical problem solving: Some underrepresented themes and needed directions. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 247–266). Hillsdale, NJ: Erlbaum.Google Scholar
  47. Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM Mathematics Education, 3, 75–80.CrossRefGoogle Scholar
  48. Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C., & Font Strawhun, B. T. (2005). Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom. Journal of Mathematical Behavior, 24, 287–301.CrossRefGoogle Scholar
  49. Silver, E. A., Leung, S. S., & Cai, J. (1995). Generating multiple solutions for a problem: A comparison of the responses of U.S. and Japanese students. Educational Studies in Mathematics, 28, 35–54.CrossRefGoogle Scholar
  50. Singer, F. M., Ellerton, N. F., & Cai, J. (2015). Mathematical problem posing: From research to effective practice. New York, NY: Springer.CrossRefGoogle Scholar
  51. Skemp, R. (1987). The psychology of learning mathematics. Mahwah, NJ: Erlbaum.Google Scholar
  52. Spiro, R. J., Feltovich, P. J., Jacobson, M. J., & Coulson, R. L. (1991). Cognitive flexibility, constructivism, and hypertext: Random access instruction for advanced knowledge acquisition in ill-structured domains. In T. Duffy & D. Jonassen (Eds.), Constructivism and the technology of instruction (pp. 57–76). Hillsdale, NJ: Erlbaum.Google Scholar
  53. Spiro, R. J., & Jehng, J. C. (1990). Cognitive flexibility and hypertext: Theory and technology for the nonlinear and multidimensional traversal of complex subject matters. In D. Nix & R. J. Spiro (Eds.), Cognition, education, and multimedia: Exploring ideas in high technology (pp. 163–205). Hillsdale, NJ: Erlbaum.Google Scholar
  54. Sriraman, B. (2009). The characteristics of mathematical creativity. ZDM, 41, 13–27.Google Scholar
  55. Star, J. R., & Rittle-Johnson, B. (2008). Flexibility in problem solving: The case of equation solving. Learning and Instruction, 18(6), 565–579.CrossRefGoogle Scholar
  56. Tabachneck, H. J., Koedinger, K. R., & Nathan, M. J. (1994). Toward a theoretical account of strategy use and sense-making in mathematics problem solving. In A. Ram & K. Eiselt (Eds.), Proceedings of the Sixteenth Annual Conference of the Cognitive Science Society (pp. 836–841). Hillsdale, NJ: Erlbaum.Google Scholar
  57. The College Board. (2011b). Retrieved from https://collegereadiness.collegeboard.org/sat.
  58. Thurston, W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 20, 161–177.CrossRefGoogle Scholar
  59. Tjoe, H. (2014). When understanding evokes appreciation: The effect of mathematics content knowledge on aesthetic predisposition. In C. Nicol, S. Oesterle, P. Liljedahl, & D. Allan. (Eds.), Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education (Vol. 5, pp. 249–256). Vancouver, BC: PME.Google Scholar
  60. Tjoe, H. (2015). Giftedness and aesthetics: Perspectives of expert mathematicians and mathematically gifted students. Gifted Child Quarterly, 59(3), 165–176.CrossRefGoogle Scholar
  61. Tjoe, H., & de la Torre, J. (2014). On recognizing proportionality: Does the ability to solve missing value proportional problems presuppose the conception of proportional reasoning? Journal of Mathematical Behavior, 33(1), 1–7.CrossRefGoogle Scholar
  62. Torrance, E. P. (1966). Torrance tests of creative thinking: Norms-technical manual. Princeton, NJ: Personal Press.Google Scholar
  63. Van Someren, M. W., Boshuizen, H. P., de Jong, T., & Reimann, P. (1998). Introduction. In M. W. van Someren, P. Reimann, H. Boshuizen, & T. de Jong (Eds.), Learning with multiple representations (pp. 1–5). Oxford, England: Elsevier Sciences.Google Scholar
  64. Vogeli, B. R. (2015). Special secondary schools for the mathematically talented: An international panorama. New York, NY: Teachers College, Columbia University.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.The Pennsylvania State UniversityReadingUSA

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